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Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Ordinary Differential Equations. Session 7
Dr. Marco A Roque Sol
11/09/2017
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Let us start by solving an m × n system of linear equations
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
where aij are given coefficients, b′ms are given right-hand side, andx ′ns are the unknowns. In this way, we can introduce new arrays ofnumbers to study the linear system
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Let us start by solving an m × n system of linear equations
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
where aij are given coefficients, b′ms are given right-hand side, andx ′ns are the unknowns. In this way, we can introduce new arrays ofnumbers to study the linear system
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Let us start by solving an m × n system of linear equations
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
where aij are given coefficients, b′ms are given right-hand side, andx ′ns are the unknowns. In this way, we can introduce new arrays ofnumbers to study the linear system
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Let us start by solving an m × n system of linear equations
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
where aij are given coefficients,
b′ms are given right-hand side, andx ′ns are the unknowns. In this way, we can introduce new arrays ofnumbers to study the linear system
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Let us start by solving an m × n system of linear equations
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
where aij are given coefficients, b′ms are given right-hand side, and
x ′ns are the unknowns. In this way, we can introduce new arrays ofnumbers to study the linear system
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Let us start by solving an m × n system of linear equations
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
where aij are given coefficients, b′ms are given right-hand side, andx ′ns are the unknowns.
In this way, we can introduce new arrays ofnumbers to study the linear system
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Let us start by solving an m × n system of linear equations
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
where aij are given coefficients, b′ms are given right-hand side, andx ′ns are the unknowns. In this way, we can introduce new arrays ofnumbers
to study the linear system
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Let us start by solving an m × n system of linear equations
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
where aij are given coefficients, b′ms are given right-hand side, andx ′ns are the unknowns. In this way, we can introduce new arrays ofnumbers to study the linear system
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Let us start by solving an m × n system of linear equations
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
where aij are given coefficients, b′ms are given right-hand side, andx ′ns are the unknowns. In this way, we can introduce new arrays ofnumbers to study the linear system
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Let us start by solving an m × n system of linear equations
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
where aij are given coefficients, b′ms are given right-hand side, andx ′ns are the unknowns. In this way, we can introduce new arrays ofnumbers to study the linear system
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Let us start by solving an m × n system of linear equations
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
where aij are given coefficients, b′ms are given right-hand side, andx ′ns are the unknowns. In this way, we can introduce new arrays ofnumbers to study the linear system
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
In this way, we have the following
Definition
An m× n matrix A , is an array of complex numbers ( m-rows andn-columns ),denoted by
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
= (aij)m×n
In this context, an element in the i-row and j-column is of thematrix A denoted by aij .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
In this way, we have the following
Definition
An m× n matrix A , is an array of complex numbers ( m-rows andn-columns ),denoted by
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
= (aij)m×n
In this context, an element in the i-row and j-column is of thematrix A denoted by aij .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
In this way, we have the following
Definition
An m× n matrix A , is an array of complex numbers ( m-rows andn-columns ),denoted by
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
= (aij)m×n
In this context, an element in the i-row and j-column is of thematrix A denoted by aij .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
In this way, we have the following
Definition
An m× n matrix A ,
is an array of complex numbers ( m-rows andn-columns ),denoted by
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
= (aij)m×n
In this context, an element in the i-row and j-column is of thematrix A denoted by aij .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
In this way, we have the following
Definition
An m× n matrix A , is an array of complex numbers ( m-rows andn-columns ),
denoted by
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
= (aij)m×n
In this context, an element in the i-row and j-column is of thematrix A denoted by aij .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
In this way, we have the following
Definition
An m× n matrix A , is an array of complex numbers ( m-rows andn-columns ),denoted by
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
= (aij)m×n
In this context, an element in the i-row and j-column is of thematrix A denoted by aij .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
In this way, we have the following
Definition
An m× n matrix A , is an array of complex numbers ( m-rows andn-columns ),denoted by
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
= (aij)m×n
In this context, an element in the i-row and j-column is of thematrix A denoted by aij .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
In this way, we have the following
Definition
An m× n matrix A , is an array of complex numbers ( m-rows andn-columns ),denoted by
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
= (aij)m×n
In this context,
an element in the i-row and j-column is of thematrix A denoted by aij .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
In this way, we have the following
Definition
An m× n matrix A , is an array of complex numbers ( m-rows andn-columns ),denoted by
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
= (aij)m×n
In this context, an element in the i-row and j-column is of thematrix A denoted by aij .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Associated with any m × n A matrix, we have the followingmatrices:a) Transpose
Is the ( n ×m ) matrix, denoted by AT , and defined by
AT =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(aTij
)n×m
= (aji )m×n
b) Complex Conjugate
Is the ( m × n ) matrix, denoted by A, and defined by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Associated with any
m × n A matrix, we have the followingmatrices:a) Transpose
Is the ( n ×m ) matrix, denoted by AT , and defined by
AT =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(aTij
)n×m
= (aji )m×n
b) Complex Conjugate
Is the ( m × n ) matrix, denoted by A, and defined by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Associated with any m × n A matrix,
we have the followingmatrices:a) Transpose
Is the ( n ×m ) matrix, denoted by AT , and defined by
AT =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(aTij
)n×m
= (aji )m×n
b) Complex Conjugate
Is the ( m × n ) matrix, denoted by A, and defined by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Associated with any m × n A matrix, we have the followingmatrices:
a) Transpose
Is the ( n ×m ) matrix, denoted by AT , and defined by
AT =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(aTij
)n×m
= (aji )m×n
b) Complex Conjugate
Is the ( m × n ) matrix, denoted by A, and defined by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Associated with any m × n A matrix, we have the followingmatrices:a) Transpose
Is the ( n ×m ) matrix, denoted by AT , and defined by
AT =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(aTij
)n×m
= (aji )m×n
b) Complex Conjugate
Is the ( m × n ) matrix, denoted by A, and defined by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Associated with any m × n A matrix, we have the followingmatrices:a) Transpose
Is the ( n ×m ) matrix, denoted by AT , and
defined by
AT =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(aTij
)n×m
= (aji )m×n
b) Complex Conjugate
Is the ( m × n ) matrix, denoted by A, and defined by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Associated with any m × n A matrix, we have the followingmatrices:a) Transpose
Is the ( n ×m ) matrix, denoted by AT , and defined by
AT =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(aTij
)n×m
= (aji )m×n
b) Complex Conjugate
Is the ( m × n ) matrix, denoted by A, and defined by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Associated with any m × n A matrix, we have the followingmatrices:a) Transpose
Is the ( n ×m ) matrix, denoted by AT , and defined by
AT =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(aTij
)n×m
= (aji )m×n
b) Complex Conjugate
Is the ( m × n ) matrix, denoted by A, and defined by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Associated with any m × n A matrix, we have the followingmatrices:a) Transpose
Is the ( n ×m ) matrix, denoted by AT , and defined by
AT =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(aTij
)n×m
= (aji )m×n
b) Complex Conjugate
Is the ( m × n ) matrix, denoted by A, and defined by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Associated with any m × n A matrix, we have the followingmatrices:a) Transpose
Is the ( n ×m ) matrix, denoted by AT , and defined by
AT =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(aTij
)n×m
= (aji )m×n
b) Complex Conjugate
Is the ( m × n ) matrix, denoted by A, and
defined by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Associated with any m × n A matrix, we have the followingmatrices:a) Transpose
Is the ( n ×m ) matrix, denoted by AT , and defined by
AT =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(aTij
)n×m
= (aji )m×n
b) Complex Conjugate
Is the ( m × n ) matrix, denoted by A, and defined by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
= (aij)m×n = (aij)m×n
c) Adjoint
Is the ( m × n ) matrix, denoted by A∗ = AT
, and defined by
A∗ =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(a∗ij)n×m = (aji )m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
= (aij)m×n = (aij)m×n
c) Adjoint
Is the ( m × n ) matrix, denoted by A∗ = AT
, and defined by
A∗ =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(a∗ij)n×m = (aji )m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
= (aij)m×n = (aij)m×n
c) Adjoint
Is the ( m × n ) matrix, denoted by A∗ = AT
, and defined by
A∗ =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(a∗ij)n×m = (aji )m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
= (aij)m×n = (aij)m×n
c) Adjoint
Is the ( m × n ) matrix, denoted by A∗ = AT
, and
defined by
A∗ =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(a∗ij)n×m = (aji )m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
= (aij)m×n = (aij)m×n
c) Adjoint
Is the ( m × n ) matrix, denoted by A∗ = AT
, and defined by
A∗ =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(a∗ij)n×m = (aji )m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
= (aij)m×n = (aij)m×n
c) Adjoint
Is the ( m × n ) matrix, denoted by A∗ = AT
, and defined by
A∗ =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(a∗ij)n×m = (aji )m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Basic Matrix Operations
Let A = (aij)m×n and B = (bij)m×n be two matrices, then wedefine
1) A = B ⇐⇒
aij = bij ; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Addtion
A± B = (aij ± bij)m×ne) Scalar Multiplication
rA = (raij)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Basic Matrix Operations
Let A = (aij)m×n and B = (bij)m×n be two matrices, then wedefine
1) A = B ⇐⇒
aij = bij ; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Addtion
A± B = (aij ± bij)m×ne) Scalar Multiplication
rA = (raij)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Basic Matrix Operations
Let A = (aij)m×n and B = (bij)m×n be two matrices, then wedefine
1) A = B ⇐⇒
aij = bij ; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Addtion
A± B = (aij ± bij)m×ne) Scalar Multiplication
rA = (raij)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Basic Matrix Operations
Let A = (aij)m×n and B = (bij)m×n be two matrices, then wedefine
1) A = B ⇐⇒
aij = bij ; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Addtion
A± B = (aij ± bij)m×ne) Scalar Multiplication
rA = (raij)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Basic Matrix Operations
Let A = (aij)m×n and B = (bij)m×n be two matrices, then wedefine
1) A = B ⇐⇒
aij = bij ; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Addtion
A± B = (aij ± bij)m×ne) Scalar Multiplication
rA = (raij)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Basic Matrix Operations
Let A = (aij)m×n and B = (bij)m×n be two matrices, then wedefine
1) A = B ⇐⇒
aij = bij ; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Addtion
A± B =
(aij ± bij)m×ne) Scalar Multiplication
rA = (raij)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Basic Matrix Operations
Let A = (aij)m×n and B = (bij)m×n be two matrices, then wedefine
1) A = B ⇐⇒
aij = bij ; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Addtion
A± B = (aij ± bij)m×n
e) Scalar Multiplication
rA = (raij)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Basic Matrix Operations
Let A = (aij)m×n and B = (bij)m×n be two matrices, then wedefine
1) A = B ⇐⇒
aij = bij ; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Addtion
A± B = (aij ± bij)m×ne) Scalar Multiplication
rA = (raij)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Basic Matrix Operations
Let A = (aij)m×n and B = (bij)m×n be two matrices, then wedefine
1) A = B ⇐⇒
aij = bij ; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Addtion
A± B = (aij ± bij)m×ne) Scalar Multiplication
rA =
(raij)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Basic Matrix Operations
Let A = (aij)m×n and B = (bij)m×n be two matrices, then wedefine
1) A = B ⇐⇒
aij = bij ; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Addtion
A± B = (aij ± bij)m×ne) Scalar Multiplication
rA = (raij)m×nDr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Multiplication
Let A and B, m × p and p × n matrices respectively
AB = (cij)m×n
where
cij =
p∑k=1
aikbkj
(AB)ij = cij =
. . . . . .. . . . . .ai1 ai2 . . . ain
. . ....
. . . b1j . . .
. . . b2j . . .
. . .... . . .
. . . bnj . . .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Multiplication
Let A and B, m × p and p × n matrices respectively
AB = (cij)m×n
where
cij =
p∑k=1
aikbkj
(AB)ij = cij =
. . . . . .. . . . . .ai1 ai2 . . . ain
. . ....
. . . b1j . . .
. . . b2j . . .
. . .... . . .
. . . bnj . . .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Multiplication
Let A and B, m × p and p × n matrices respectively
AB = (cij)m×n
where
cij =
p∑k=1
aikbkj
(AB)ij = cij =
. . . . . .. . . . . .ai1 ai2 . . . ain
. . ....
. . . b1j . . .
. . . b2j . . .
. . .... . . .
. . . bnj . . .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Multiplication
Let A and B, m × p and p × n matrices respectively
AB = (cij)m×n
where
cij =
p∑k=1
aikbkj
(AB)ij = cij =
. . . . . .. . . . . .ai1 ai2 . . . ain
. . ....
. . . b1j . . .
. . . b2j . . .
. . .... . . .
. . . bnj . . .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Multiplication
Let A and B, m × p and p × n matrices respectively
AB = (cij)m×n
where
cij =
p∑k=1
aikbkj
(AB)ij = cij =
. . . . . .. . . . . .ai1 ai2 . . . ain
. . ....
. . . b1j . . .
. . . b2j . . .
. . .... . . .
. . . bnj . . .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Multiplication
Let A and B, m × p and p × n matrices respectively
AB = (cij)m×n
where
cij =
p∑k=1
aikbkj
(AB)ij = cij =
. . . . . .. . . . . .ai1 ai2 . . . ain
. . ....
. . . b1j . . .
. . . b2j . . .
. . .... . . .
. . . bnj . . .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Multiplication
Let A and B, m × p and p × n matrices respectively
AB = (cij)m×n
where
cij =
p∑k=1
aikbkj
(AB)ij = cij =
. . . . . .. . . . . .ai1 ai2 . . . ain
. . ....
. . . b1j . . .
. . . b2j . . .
. . .... . . .
. . . bnj . . .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
In general, when AB is defined, not necessarily BA is also defined,but even in that case, we have in general
AB 6= BA
Example 7.1
Let A and B the matrices defined by
A =
1 −2 10 2 −12 1 1
B =
2 1 −11 −1 02 −1 1
Find A + B, A− B, 3A AB, BA
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
In general, when AB is defined, not necessarily BA is also defined,but even in that case, we have in general
AB 6= BA
Example 7.1
Let A and B the matrices defined by
A =
1 −2 10 2 −12 1 1
B =
2 1 −11 −1 02 −1 1
Find A + B, A− B, 3A AB, BA
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
In general, when AB is defined, not necessarily BA is also defined,
but even in that case, we have in general
AB 6= BA
Example 7.1
Let A and B the matrices defined by
A =
1 −2 10 2 −12 1 1
B =
2 1 −11 −1 02 −1 1
Find A + B, A− B, 3A AB, BA
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
In general, when AB is defined, not necessarily BA is also defined,but even in that case, we have in general
AB 6= BA
Example 7.1
Let A and B the matrices defined by
A =
1 −2 10 2 −12 1 1
B =
2 1 −11 −1 02 −1 1
Find A + B, A− B, 3A AB, BA
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
In general, when AB is defined, not necessarily BA is also defined,but even in that case, we have in general
AB 6= BA
Example 7.1
Let A and B the matrices defined by
A =
1 −2 10 2 −12 1 1
B =
2 1 −11 −1 02 −1 1
Find A + B, A− B, 3A AB, BA
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
In general, when AB is defined, not necessarily BA is also defined,but even in that case, we have in general
AB 6= BA
Example 7.1
Let A and B the matrices defined by
A =
1 −2 10 2 −12 1 1
B =
2 1 −11 −1 02 −1 1
Find A + B, A− B, 3A AB, BA
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
In general, when AB is defined, not necessarily BA is also defined,but even in that case, we have in general
AB 6= BA
Example 7.1
Let A and B the matrices defined by
A =
1 −2 10 2 −12 1 1
B =
2 1 −11 −1 02 −1 1
Find A + B, A− B, 3A AB, BA
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
In general, when AB is defined, not necessarily BA is also defined,but even in that case, we have in general
AB 6= BA
Example 7.1
Let A and B the matrices defined by
A =
1 −2 10 2 −12 1 1
B =
2 1 −11 −1 02 −1 1
Find A + B, A− B, 3A AB, BA
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Solution
A + B =
3 −1 01 1 −14 0 2
A− B =
−1 −3 2−1 3 −10 2 0
3A =
6 3 −3−3 3 06 −3 3
AB =
1 −2 10 2 −12 1 1
2 1 −11 −1 02 −1 1
=
2 2 00 −1 −17 0 −1
BA =
2 1 −11 −1 02 −1 1
1 −2 10 2 −12 1 1
=
0 −3 01 0 24 −5 4
6= AB
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Solution
A + B =
3 −1 01 1 −14 0 2
A− B =
−1 −3 2−1 3 −10 2 0
3A =
6 3 −3−3 3 06 −3 3
AB =
1 −2 10 2 −12 1 1
2 1 −11 −1 02 −1 1
=
2 2 00 −1 −17 0 −1
BA =
2 1 −11 −1 02 −1 1
1 −2 10 2 −12 1 1
=
0 −3 01 0 24 −5 4
6= AB
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Solution
A + B =
3 −1 01 1 −14 0 2
A− B =
−1 −3 2−1 3 −10 2 0
3A =
6 3 −3−3 3 06 −3 3
AB =
1 −2 10 2 −12 1 1
2 1 −11 −1 02 −1 1
=
2 2 00 −1 −17 0 −1
BA =
2 1 −11 −1 02 −1 1
1 −2 10 2 −12 1 1
=
0 −3 01 0 24 −5 4
6= AB
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Solution
A + B =
3 −1 01 1 −14 0 2
A− B =
−1 −3 2−1 3 −10 2 0
3A =
6 3 −3−3 3 06 −3 3
AB =
1 −2 10 2 −12 1 1
2 1 −11 −1 02 −1 1
=
2 2 00 −1 −17 0 −1
BA =
2 1 −11 −1 02 −1 1
1 −2 10 2 −12 1 1
=
0 −3 01 0 24 −5 4
6= AB
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Solution
A + B =
3 −1 01 1 −14 0 2
A− B =
−1 −3 2−1 3 −10 2 0
3A =
6 3 −3−3 3 06 −3 3
AB =
1 −2 10 2 −12 1 1
2 1 −11 −1 02 −1 1
=
2 2 00 −1 −17 0 −1
BA =
2 1 −11 −1 02 −1 1
1 −2 10 2 −12 1 1
=
0 −3 01 0 24 −5 4
6= AB
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Solution
A + B =
3 −1 01 1 −14 0 2
A− B =
−1 −3 2−1 3 −10 2 0
3A =
6 3 −3−3 3 06 −3 3
AB =
1 −2 10 2 −12 1 1
2 1 −11 −1 02 −1 1
=
2 2 00 −1 −17 0 −1
BA =
2 1 −11 −1 02 −1 1
1 −2 10 2 −12 1 1
=
0 −3 01 0 24 −5 4
6= AB
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Solution
A + B =
3 −1 01 1 −14 0 2
A− B =
−1 −3 2−1 3 −10 2 0
3A =
6 3 −3−3 3 06 −3 3
AB =
1 −2 10 2 −12 1 1
2 1 −11 −1 02 −1 1
=
2 2 00 −1 −17 0 −1
BA =
2 1 −11 −1 02 −1 1
1 −2 10 2 −12 1 1
=
0 −3 01 0 24 −5 4
6= AB
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Solution
A + B =
3 −1 01 1 −14 0 2
A− B =
−1 −3 2−1 3 −10 2 0
3A =
6 3 −3−3 3 06 −3 3
AB =
1 −2 10 2 −12 1 1
2 1 −11 −1 02 −1 1
=
2 2 00 −1 −17 0 −1
BA =
2 1 −11 −1 02 −1 1
1 −2 10 2 −12 1 1
=
0 −3 01 0 24 −5 4
6= AB
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Solution
A + B =
3 −1 01 1 −14 0 2
A− B =
−1 −3 2−1 3 −10 2 0
3A =
6 3 −3−3 3 06 −3 3
AB =
1 −2 10 2 −12 1 1
2 1 −11 −1 02 −1 1
=
2 2 00 −1 −17 0 −1
BA =
2 1 −11 −1 02 −1 1
1 −2 10 2 −12 1 1
=
0 −3 01 0 24 −5 4
6= AB
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Solution
A + B =
3 −1 01 1 −14 0 2
A− B =
−1 −3 2−1 3 −10 2 0
3A =
6 3 −3−3 3 06 −3 3
AB =
1 −2 10 2 −12 1 1
2 1 −11 −1 02 −1 1
=
2 2 00 −1 −17 0 −1
BA =
2 1 −11 −1 02 −1 1
1 −2 10 2 −12 1 1
=
0 −3 01 0 24 −5 4
6= AB
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Solution
A + B =
3 −1 01 1 −14 0 2
A− B =
−1 −3 2−1 3 −10 2 0
3A =
6 3 −3−3 3 06 −3 3
AB =
1 −2 10 2 −12 1 1
2 1 −11 −1 02 −1 1
=
2 2 00 −1 −17 0 −1
BA =
2 1 −11 −1 02 −1 1
1 −2 10 2 −12 1 1
=
0 −3 01 0 24 −5 4
6= AB
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Solution
A + B =
3 −1 01 1 −14 0 2
A− B =
−1 −3 2−1 3 −10 2 0
3A =
6 3 −3−3 3 06 −3 3
AB =
1 −2 10 2 −12 1 1
2 1 −11 −1 02 −1 1
=
2 2 00 −1 −17 0 −1
BA =
2 1 −11 −1 02 −1 1
1 −2 10 2 −12 1 1
=
0 −3 01 0 24 −5 4
6= AB
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Solution
A + B =
3 −1 01 1 −14 0 2
A− B =
−1 −3 2−1 3 −10 2 0
3A =
6 3 −3−3 3 06 −3 3
AB =
1 −2 10 2 −12 1 1
2 1 −11 −1 02 −1 1
=
2 2 00 −1 −17 0 −1
BA =
2 1 −11 −1 02 −1 1
1 −2 10 2 −12 1 1
=
0 −3 01 0 24 −5 4
6= AB
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Solution
A + B =
3 −1 01 1 −14 0 2
A− B =
−1 −3 2−1 3 −10 2 0
3A =
6 3 −3−3 3 06 −3 3
AB =
1 −2 10 2 −12 1 1
2 1 −11 −1 02 −1 1
=
2 2 00 −1 −17 0 −1
BA =
2 1 −11 −1 02 −1 1
1 −2 10 2 −12 1 1
=
0 −3 01 0 24 −5 4
6= AB
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.2
Let C and D the matrices defined by
C =
2 11 −12 −1
D =
(1 −2 10 2 −1
)Find CD and DC.Solution
CD =
2 11 −12 −1
(1 −2 10 2 −1
)=
2 −2 11 −4 22 −6 3
DC =
(1 −2 10 2 −1
) 2 11 −12 −1
=
(2 20 −1
)6= CD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.2
Let C and D the matrices defined by
C =
2 11 −12 −1
D =
(1 −2 10 2 −1
)Find CD and DC.Solution
CD =
2 11 −12 −1
(1 −2 10 2 −1
)=
2 −2 11 −4 22 −6 3
DC =
(1 −2 10 2 −1
) 2 11 −12 −1
=
(2 20 −1
)6= CD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.2
Let C and D the matrices defined by
C =
2 11 −12 −1
D =
(1 −2 10 2 −1
)Find CD and DC.Solution
CD =
2 11 −12 −1
(1 −2 10 2 −1
)=
2 −2 11 −4 22 −6 3
DC =
(1 −2 10 2 −1
) 2 11 −12 −1
=
(2 20 −1
)6= CD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.2
Let C and D the matrices defined by
C =
2 11 −12 −1
D =
(1 −2 10 2 −1
)Find CD and DC.Solution
CD =
2 11 −12 −1
(1 −2 10 2 −1
)=
2 −2 11 −4 22 −6 3
DC =
(1 −2 10 2 −1
) 2 11 −12 −1
=
(2 20 −1
)6= CD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.2
Let C and D the matrices defined by
C =
2 11 −12 −1
D =
(1 −2 10 2 −1
)
Find CD and DC.Solution
CD =
2 11 −12 −1
(1 −2 10 2 −1
)=
2 −2 11 −4 22 −6 3
DC =
(1 −2 10 2 −1
) 2 11 −12 −1
=
(2 20 −1
)6= CD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.2
Let C and D the matrices defined by
C =
2 11 −12 −1
D =
(1 −2 10 2 −1
)Find CD and DC.
Solution
CD =
2 11 −12 −1
(1 −2 10 2 −1
)=
2 −2 11 −4 22 −6 3
DC =
(1 −2 10 2 −1
) 2 11 −12 −1
=
(2 20 −1
)6= CD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.2
Let C and D the matrices defined by
C =
2 11 −12 −1
D =
(1 −2 10 2 −1
)Find CD and DC.Solution
CD =
2 11 −12 −1
(1 −2 10 2 −1
)=
2 −2 11 −4 22 −6 3
DC =
(1 −2 10 2 −1
) 2 11 −12 −1
=
(2 20 −1
)6= CD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.2
Let C and D the matrices defined by
C =
2 11 −12 −1
D =
(1 −2 10 2 −1
)Find CD and DC.Solution
CD =
2 11 −12 −1
(1 −2 10 2 −1
)=
2 −2 11 −4 22 −6 3
DC =
(1 −2 10 2 −1
) 2 11 −12 −1
=
(2 20 −1
)6= CD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.2
Let C and D the matrices defined by
C =
2 11 −12 −1
D =
(1 −2 10 2 −1
)Find CD and DC.Solution
CD =
2 11 −12 −1
(1 −2 10 2 −1
)=
2 −2 11 −4 22 −6 3
DC =
(1 −2 10 2 −1
) 2 11 −12 −1
=
(2 20 −1
)6= CD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.2
Let C and D the matrices defined by
C =
2 11 −12 −1
D =
(1 −2 10 2 −1
)Find CD and DC.Solution
CD =
2 11 −12 −1
(1 −2 10 2 −1
)=
2 −2 11 −4 22 −6 3
DC =
(1 −2 10 2 −1
) 2 11 −12 −1
=
(2 20 −1
)6= CD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.2
Let C and D the matrices defined by
C =
2 11 −12 −1
D =
(1 −2 10 2 −1
)Find CD and DC.Solution
CD =
2 11 −12 −1
(1 −2 10 2 −1
)=
2 −2 11 −4 22 −6 3
DC =
(1 −2 10 2 −1
) 2 11 −12 −1
=
(2 20 −1
)6= CD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.2
Let C and D the matrices defined by
C =
2 11 −12 −1
D =
(1 −2 10 2 −1
)Find CD and DC.Solution
CD =
2 11 −12 −1
(1 −2 10 2 −1
)=
2 −2 11 −4 22 −6 3
DC =
(1 −2 10 2 −1
) 2 11 −12 −1
=
(2 20 −1
)6= CD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.2
Let C and D the matrices defined by
C =
2 11 −12 −1
D =
(1 −2 10 2 −1
)Find CD and DC.Solution
CD =
2 11 −12 −1
(1 −2 10 2 −1
)=
2 −2 11 −4 22 −6 3
DC =
(1 −2 10 2 −1
)
2 11 −12 −1
=
(2 20 −1
)6= CD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.2
Let C and D the matrices defined by
C =
2 11 −12 −1
D =
(1 −2 10 2 −1
)Find CD and DC.Solution
CD =
2 11 −12 −1
(1 −2 10 2 −1
)=
2 −2 11 −4 22 −6 3
DC =
(1 −2 10 2 −1
) 2 11 −12 −1
=
(2 20 −1
)6= CD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.2
Let C and D the matrices defined by
C =
2 11 −12 −1
D =
(1 −2 10 2 −1
)Find CD and DC.Solution
CD =
2 11 −12 −1
(1 −2 10 2 −1
)=
2 −2 11 −4 22 −6 3
DC =
(1 −2 10 2 −1
) 2 11 −12 −1
=
(2 20 −1
)6= CD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.3
Using matrix operations rewrite the linear system
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
in terms of matrices.
Solution
Starting with the system
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.3
Using matrix operations rewrite the linear system
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
in terms of matrices.
Solution
Starting with the system
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.3
Using matrix operations rewrite the linear system
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
in terms of matrices.
Solution
Starting with the system
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.3
Using matrix operations rewrite the linear system
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
in terms of matrices.
Solution
Starting with the system
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.3
Using matrix operations rewrite the linear system
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
in terms of matrices.
Solution
Starting with the system
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.3
Using matrix operations rewrite the linear system
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
in terms of matrices.
Solution
Starting with the system
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
and choosing
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
and choosing
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
and choosing
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
and choosing
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
and choosing
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
and choosing
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
and choosing
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
AX =
a11x1 + a12x2 + . . .+ a1nxna21x1 + a22x2 + . . .+ a2nxn
...am1x1 + am2x2 + . . .+ amnxn
=
b1b2...bm
= B =⇒ AX = B
Types of Matrices An m × n matrix A = (aij)m×n is a
1) Zero matrix if aij = 0; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Square Matrix if m = n.
A =
2 −2 11 −4 22 −6 3
; B
(3 75 −4
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
AX =
a11x1 + a12x2 + . . .+ a1nxna21x1 + a22x2 + . . .+ a2nxn
...am1x1 + am2x2 + . . .+ amnxn
=
b1b2...bm
= B =⇒ AX = B
Types of Matrices An m × n matrix A = (aij)m×n is a
1) Zero matrix if aij = 0; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Square Matrix if m = n.
A =
2 −2 11 −4 22 −6 3
; B
(3 75 −4
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
AX =
a11x1 + a12x2 + . . .+ a1nxna21x1 + a22x2 + . . .+ a2nxn
...am1x1 + am2x2 + . . .+ amnxn
=
b1b2...bm
= B =⇒
AX = B
Types of Matrices An m × n matrix A = (aij)m×n is a
1) Zero matrix if aij = 0; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Square Matrix if m = n.
A =
2 −2 11 −4 22 −6 3
; B
(3 75 −4
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
AX =
a11x1 + a12x2 + . . .+ a1nxna21x1 + a22x2 + . . .+ a2nxn
...am1x1 + am2x2 + . . .+ amnxn
=
b1b2...bm
= B =⇒ AX = B
Types of Matrices An m × n matrix A = (aij)m×n is a
1) Zero matrix if aij = 0; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Square Matrix if m = n.
A =
2 −2 11 −4 22 −6 3
; B
(3 75 −4
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
AX =
a11x1 + a12x2 + . . .+ a1nxna21x1 + a22x2 + . . .+ a2nxn
...am1x1 + am2x2 + . . .+ amnxn
=
b1b2...bm
= B =⇒ AX = B
Types of Matrices
An m × n matrix A = (aij)m×n is a
1) Zero matrix if aij = 0; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Square Matrix if m = n.
A =
2 −2 11 −4 22 −6 3
; B
(3 75 −4
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
AX =
a11x1 + a12x2 + . . .+ a1nxna21x1 + a22x2 + . . .+ a2nxn
...am1x1 + am2x2 + . . .+ amnxn
=
b1b2...bm
= B =⇒ AX = B
Types of Matrices An m × n matrix A = (aij)m×n is a
1) Zero matrix if aij = 0; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Square Matrix if m = n.
A =
2 −2 11 −4 22 −6 3
; B
(3 75 −4
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
AX =
a11x1 + a12x2 + . . .+ a1nxna21x1 + a22x2 + . . .+ a2nxn
...am1x1 + am2x2 + . . .+ amnxn
=
b1b2...bm
= B =⇒ AX = B
Types of Matrices An m × n matrix A = (aij)m×n is a
1) Zero matrix if aij = 0; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Square Matrix if m = n.
A =
2 −2 11 −4 22 −6 3
; B
(3 75 −4
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
AX =
a11x1 + a12x2 + . . .+ a1nxna21x1 + a22x2 + . . .+ a2nxn
...am1x1 + am2x2 + . . .+ amnxn
=
b1b2...bm
= B =⇒ AX = B
Types of Matrices An m × n matrix A = (aij)m×n is a
1) Zero matrix if aij = 0; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Square Matrix if m = n.
A =
2 −2 11 −4 22 −6 3
; B
(3 75 −4
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
AX =
a11x1 + a12x2 + . . .+ a1nxna21x1 + a22x2 + . . .+ a2nxn
...am1x1 + am2x2 + . . .+ amnxn
=
b1b2...bm
= B =⇒ AX = B
Types of Matrices An m × n matrix A = (aij)m×n is a
1) Zero matrix if aij = 0; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Square Matrix if m = n.
A =
2 −2 11 −4 22 −6 3
;
B
(3 75 −4
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
AX =
a11x1 + a12x2 + . . .+ a1nxna21x1 + a22x2 + . . .+ a2nxn
...am1x1 + am2x2 + . . .+ amnxn
=
b1b2...bm
= B =⇒ AX = B
Types of Matrices An m × n matrix A = (aij)m×n is a
1) Zero matrix if aij = 0; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Square Matrix if m = n.
A =
2 −2 11 −4 22 −6 3
; B
(3 75 −4
)Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
3) Identity matrix (n × n) (I) if aij = δij where δij =
{1 i = j0 i 6= j
A = I =
1 0
1. . .
0 1
4) Symetric Matrix (n × n) if AT = A or aij = aji ;
i = 1, 2, ...,m, j = 1, 2, ..., n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
3) Identity matrix (n × n)
(I) if aij = δij where δij =
{1 i = j0 i 6= j
A = I =
1 0
1. . .
0 1
4) Symetric Matrix (n × n) if AT = A or aij = aji ;
i = 1, 2, ...,m, j = 1, 2, ..., n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
3) Identity matrix (n × n) (I)
if aij = δij where δij =
{1 i = j0 i 6= j
A = I =
1 0
1. . .
0 1
4) Symetric Matrix (n × n) if AT = A or aij = aji ;
i = 1, 2, ...,m, j = 1, 2, ..., n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
3) Identity matrix (n × n) (I) if aij = δij
where δij =
{1 i = j0 i 6= j
A = I =
1 0
1. . .
0 1
4) Symetric Matrix (n × n) if AT = A or aij = aji ;
i = 1, 2, ...,m, j = 1, 2, ..., n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
3) Identity matrix (n × n) (I) if aij = δij where δij =
{1 i = j0 i 6= j
A = I =
1 0
1. . .
0 1
4) Symetric Matrix (n × n) if AT = A or aij = aji ;
i = 1, 2, ...,m, j = 1, 2, ..., n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
3) Identity matrix (n × n) (I) if aij = δij where δij =
{1 i = j0 i 6= j
A =
I =
1 0
1. . .
0 1
4) Symetric Matrix (n × n) if AT = A or aij = aji ;
i = 1, 2, ...,m, j = 1, 2, ..., n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
3) Identity matrix (n × n) (I) if aij = δij where δij =
{1 i = j0 i 6= j
A = I =
1 0
1. . .
0 1
4) Symetric Matrix (n × n) if AT = A or aij = aji ;
i = 1, 2, ...,m, j = 1, 2, ..., n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
3) Identity matrix (n × n) (I) if aij = δij where δij =
{1 i = j0 i 6= j
A = I =
1 0
1. . .
0 1
4) Symetric Matrix (n × n)
if AT = A or aij = aji ;i = 1, 2, ...,m, j = 1, 2, ..., n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
3) Identity matrix (n × n) (I) if aij = δij where δij =
{1 i = j0 i 6= j
A = I =
1 0
1. . .
0 1
4) Symetric Matrix (n × n) if AT = A or aij = aji ;
i = 1, 2, ...,m, j = 1, 2, ..., n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
5) Triangular Matrix (n × n)
5a) Upper Triangular Matrix(U) if uij = 0, i > j
U =
a11 · · · · · · a1n
a22. . .
...
0 ann
5b) Lower Triangular Matrix(L) if lij = 0, i < j
L =
a11
a22 0. . .
... · · · · · · ann
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
5) Triangular Matrix (n × n)
5a) Upper Triangular Matrix(U) if uij = 0, i > j
U =
a11 · · · · · · a1n
a22. . .
...
0 ann
5b) Lower Triangular Matrix(L) if lij = 0, i < j
L =
a11
a22 0. . .
... · · · · · · ann
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
5) Triangular Matrix (n × n)
5a) Upper Triangular Matrix
(U) if uij = 0, i > j
U =
a11 · · · · · · a1n
a22. . .
...
0 ann
5b) Lower Triangular Matrix(L) if lij = 0, i < j
L =
a11
a22 0. . .
... · · · · · · ann
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
5) Triangular Matrix (n × n)
5a) Upper Triangular Matrix(U)
if uij = 0, i > j
U =
a11 · · · · · · a1n
a22. . .
...
0 ann
5b) Lower Triangular Matrix(L) if lij = 0, i < j
L =
a11
a22 0. . .
... · · · · · · ann
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
5) Triangular Matrix (n × n)
5a) Upper Triangular Matrix(U) if uij = 0, i > j
U =
a11 · · · · · · a1n
a22. . .
...
0 ann
5b) Lower Triangular Matrix(L) if lij = 0, i < j
L =
a11
a22 0. . .
... · · · · · · ann
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
5) Triangular Matrix (n × n)
5a) Upper Triangular Matrix(U) if uij = 0, i > j
U =
a11 · · · · · · a1n
a22. . .
...
0 ann
5b) Lower Triangular Matrix(L) if lij = 0, i < j
L =
a11
a22 0. . .
... · · · · · · ann
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
5) Triangular Matrix (n × n)
5a) Upper Triangular Matrix(U) if uij = 0, i > j
U =
a11 · · · · · · a1n
a22. . .
...
0 ann
5b) Lower Triangular Matrix(L) if lij = 0, i < j
L =
a11
a22 0. . .
... · · · · · · ann
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
5) Triangular Matrix (n × n)
5a) Upper Triangular Matrix(U) if uij = 0, i > j
U =
a11 · · · · · · a1n
a22. . .
...
0 ann
5b) Lower Triangular Matrix
(L) if lij = 0, i < j
L =
a11
a22 0. . .
... · · · · · · ann
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
5) Triangular Matrix (n × n)
5a) Upper Triangular Matrix(U) if uij = 0, i > j
U =
a11 · · · · · · a1n
a22. . .
...
0 ann
5b) Lower Triangular Matrix(L)
if lij = 0, i < j
L =
a11
a22 0. . .
... · · · · · · ann
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
5) Triangular Matrix (n × n)
5a) Upper Triangular Matrix(U) if uij = 0, i > j
U =
a11 · · · · · · a1n
a22. . .
...
0 ann
5b) Lower Triangular Matrix(L) if lij = 0, i < j
L =
a11
a22 0. . .
... · · · · · · ann
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
5) Triangular Matrix (n × n)
5a) Upper Triangular Matrix(U) if uij = 0, i > j
U =
a11 · · · · · · a1n
a22. . .
...
0 ann
5b) Lower Triangular Matrix(L) if lij = 0, i < j
L =
a11
a22 0. . .
... · · · · · · ann
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
5) Triangular Matrix (n × n)
5a) Upper Triangular Matrix(U) if uij = 0, i > j
U =
a11 · · · · · · a1n
a22. . .
...
0 ann
5b) Lower Triangular Matrix(L) if lij = 0, i < j
L =
a11
a22 0. . .
... · · · · · · ann
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
6) Diagonal Matrix (n × n) (D) if aij = dij where dij = Diδij
D =
a11 · · · · · ·
...
a22 00 . . .
... · · · · · · ann
7) Invertible Matrix (n × n) If A is a square matrix (n × n) andthere exists an n × n matrix B such that
AB = BA = I
The matrix B is denoted by A−1 and is called the Inverse Matrixand A is called invertible or nonsingular matrix. Matrices that donot have an inverse are called it singular or noninvertible.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
6) Diagonal Matrix (n × n)
(D) if aij = dij where dij = Diδij
D =
a11 · · · · · ·
...
a22 00 . . .
... · · · · · · ann
7) Invertible Matrix (n × n) If A is a square matrix (n × n) andthere exists an n × n matrix B such that
AB = BA = I
The matrix B is denoted by A−1 and is called the Inverse Matrixand A is called invertible or nonsingular matrix. Matrices that donot have an inverse are called it singular or noninvertible.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
6) Diagonal Matrix (n × n) (D)
if aij = dij where dij = Diδij
D =
a11 · · · · · ·
...
a22 00 . . .
... · · · · · · ann
7) Invertible Matrix (n × n) If A is a square matrix (n × n) andthere exists an n × n matrix B such that
AB = BA = I
The matrix B is denoted by A−1 and is called the Inverse Matrixand A is called invertible or nonsingular matrix. Matrices that donot have an inverse are called it singular or noninvertible.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
6) Diagonal Matrix (n × n) (D) if aij = dij where
dij = Diδij
D =
a11 · · · · · ·
...
a22 00 . . .
... · · · · · · ann
7) Invertible Matrix (n × n) If A is a square matrix (n × n) andthere exists an n × n matrix B such that
AB = BA = I
The matrix B is denoted by A−1 and is called the Inverse Matrixand A is called invertible or nonsingular matrix. Matrices that donot have an inverse are called it singular or noninvertible.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
6) Diagonal Matrix (n × n) (D) if aij = dij where dij = Diδij
D =
a11 · · · · · ·
...
a22 00 . . .
... · · · · · · ann
7) Invertible Matrix (n × n) If A is a square matrix (n × n) andthere exists an n × n matrix B such that
AB = BA = I
The matrix B is denoted by A−1 and is called the Inverse Matrixand A is called invertible or nonsingular matrix. Matrices that donot have an inverse are called it singular or noninvertible.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
6) Diagonal Matrix (n × n) (D) if aij = dij where dij = Diδij
D =
a11 · · · · · ·
...
a22 00 . . .
... · · · · · · ann
7) Invertible Matrix (n × n) If A is a square matrix (n × n) andthere exists an n × n matrix B such that
AB = BA = I
The matrix B is denoted by A−1 and is called the Inverse Matrixand A is called invertible or nonsingular matrix. Matrices that donot have an inverse are called it singular or noninvertible.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
6) Diagonal Matrix (n × n) (D) if aij = dij where dij = Diδij
D =
a11 · · · · · ·
...
a22 00 . . .
... · · · · · · ann
7) Invertible Matrix (n × n)
If A is a square matrix (n × n) andthere exists an n × n matrix B such that
AB = BA = I
The matrix B is denoted by A−1 and is called the Inverse Matrixand A is called invertible or nonsingular matrix. Matrices that donot have an inverse are called it singular or noninvertible.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
6) Diagonal Matrix (n × n) (D) if aij = dij where dij = Diδij
D =
a11 · · · · · ·
...
a22 00 . . .
... · · · · · · ann
7) Invertible Matrix (n × n) If A is a square matrix (n × n) and
there exists an n × n matrix B such that
AB = BA = I
The matrix B is denoted by A−1 and is called the Inverse Matrixand A is called invertible or nonsingular matrix. Matrices that donot have an inverse are called it singular or noninvertible.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
6) Diagonal Matrix (n × n) (D) if aij = dij where dij = Diδij
D =
a11 · · · · · ·
...
a22 00 . . .
... · · · · · · ann
7) Invertible Matrix (n × n) If A is a square matrix (n × n) andthere exists an n × n matrix B such that
AB = BA = I
The matrix B is denoted by A−1 and is called the Inverse Matrixand A is called invertible or nonsingular matrix. Matrices that donot have an inverse are called it singular or noninvertible.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
6) Diagonal Matrix (n × n) (D) if aij = dij where dij = Diδij
D =
a11 · · · · · ·
...
a22 00 . . .
... · · · · · · ann
7) Invertible Matrix (n × n) If A is a square matrix (n × n) andthere exists an n × n matrix B such that
AB =
BA = I
The matrix B is denoted by A−1 and is called the Inverse Matrixand A is called invertible or nonsingular matrix. Matrices that donot have an inverse are called it singular or noninvertible.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
6) Diagonal Matrix (n × n) (D) if aij = dij where dij = Diδij
D =
a11 · · · · · ·
...
a22 00 . . .
... · · · · · · ann
7) Invertible Matrix (n × n) If A is a square matrix (n × n) andthere exists an n × n matrix B such that
AB = BA =
I
The matrix B is denoted by A−1 and is called the Inverse Matrixand A is called invertible or nonsingular matrix. Matrices that donot have an inverse are called it singular or noninvertible.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
6) Diagonal Matrix (n × n) (D) if aij = dij where dij = Diδij
D =
a11 · · · · · ·
...
a22 00 . . .
... · · · · · · ann
7) Invertible Matrix (n × n) If A is a square matrix (n × n) andthere exists an n × n matrix B such that
AB = BA = I
The matrix B is denoted by A−1 and is called the Inverse Matrixand A is called invertible or nonsingular matrix. Matrices that donot have an inverse are called it singular or noninvertible.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
6) Diagonal Matrix (n × n) (D) if aij = dij where dij = Diδij
D =
a11 · · · · · ·
...
a22 00 . . .
... · · · · · · ann
7) Invertible Matrix (n × n) If A is a square matrix (n × n) andthere exists an n × n matrix B such that
AB = BA = I
The matrix B
is denoted by A−1 and is called the Inverse Matrixand A is called invertible or nonsingular matrix. Matrices that donot have an inverse are called it singular or noninvertible.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
6) Diagonal Matrix (n × n) (D) if aij = dij where dij = Diδij
D =
a11 · · · · · ·
...
a22 00 . . .
... · · · · · · ann
7) Invertible Matrix (n × n) If A is a square matrix (n × n) andthere exists an n × n matrix B such that
AB = BA = I
The matrix B is denoted by
A−1 and is called the Inverse Matrixand A is called invertible or nonsingular matrix. Matrices that donot have an inverse are called it singular or noninvertible.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
6) Diagonal Matrix (n × n) (D) if aij = dij where dij = Diδij
D =
a11 · · · · · ·
...
a22 00 . . .
... · · · · · · ann
7) Invertible Matrix (n × n) If A is a square matrix (n × n) andthere exists an n × n matrix B such that
AB = BA = I
The matrix B is denoted by A−1 and
is called the Inverse Matrixand A is called invertible or nonsingular matrix. Matrices that donot have an inverse are called it singular or noninvertible.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
6) Diagonal Matrix (n × n) (D) if aij = dij where dij = Diδij
D =
a11 · · · · · ·
...
a22 00 . . .
... · · · · · · ann
7) Invertible Matrix (n × n) If A is a square matrix (n × n) andthere exists an n × n matrix B such that
AB = BA = I
The matrix B is denoted by A−1 and is called the Inverse Matrixand
A is called invertible or nonsingular matrix. Matrices that donot have an inverse are called it singular or noninvertible.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
6) Diagonal Matrix (n × n) (D) if aij = dij where dij = Diδij
D =
a11 · · · · · ·
...
a22 00 . . .
... · · · · · · ann
7) Invertible Matrix (n × n) If A is a square matrix (n × n) andthere exists an n × n matrix B such that
AB = BA = I
The matrix B is denoted by A−1 and is called the Inverse Matrixand A is called invertible or nonsingular matrix.
Matrices that donot have an inverse are called it singular or noninvertible.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
6) Diagonal Matrix (n × n) (D) if aij = dij where dij = Diδij
D =
a11 · · · · · ·
...
a22 00 . . .
... · · · · · · ann
7) Invertible Matrix (n × n) If A is a square matrix (n × n) andthere exists an n × n matrix B such that
AB = BA = I
The matrix B is denoted by A−1 and is called the Inverse Matrixand A is called invertible or nonsingular matrix. Matrices that donot have an inverse are called it singular or noninvertible.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
A−1 is the notation for the inverse of A, but keep in mind that
A−1 6= 1
A
There are various ways to compute A−1 from A, assuming that itexists. One way is the cofactor expansion .
Associated with each element aij of a given matrix is the minor Mij
from, which is the determinant of the matrix obtained by deletingthe ith row and jth column of the original matrix that is, the rowand column containing aij . Also associated with each element aij isthe cofactor Cij defined by the equation
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
A−1 is the notation for the inverse of A, but keep in mind that
A−1 6= 1
A
There are various ways to compute A−1 from A, assuming that itexists. One way is the cofactor expansion .
Associated with each element aij of a given matrix is the minor Mij
from, which is the determinant of the matrix obtained by deletingthe ith row and jth column of the original matrix that is, the rowand column containing aij . Also associated with each element aij isthe cofactor Cij defined by the equation
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
A−1 is the notation for the inverse of A,
but keep in mind that
A−1 6= 1
A
There are various ways to compute A−1 from A, assuming that itexists. One way is the cofactor expansion .
Associated with each element aij of a given matrix is the minor Mij
from, which is the determinant of the matrix obtained by deletingthe ith row and jth column of the original matrix that is, the rowand column containing aij . Also associated with each element aij isthe cofactor Cij defined by the equation
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
A−1 is the notation for the inverse of A, but keep in mind that
A−1 6= 1
A
There are various ways to compute A−1 from A, assuming that itexists. One way is the cofactor expansion .
Associated with each element aij of a given matrix is the minor Mij
from, which is the determinant of the matrix obtained by deletingthe ith row and jth column of the original matrix that is, the rowand column containing aij . Also associated with each element aij isthe cofactor Cij defined by the equation
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
A−1 is the notation for the inverse of A, but keep in mind that
A−1 6= 1
A
There are various ways to compute A−1 from A, assuming that itexists. One way is the cofactor expansion .
Associated with each element aij of a given matrix is the minor Mij
from, which is the determinant of the matrix obtained by deletingthe ith row and jth column of the original matrix that is, the rowand column containing aij . Also associated with each element aij isthe cofactor Cij defined by the equation
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
A−1 is the notation for the inverse of A, but keep in mind that
A−1 6= 1
A
There are various ways to compute A−1 from A, assuming that itexists.
One way is the cofactor expansion .
Associated with each element aij of a given matrix is the minor Mij
from, which is the determinant of the matrix obtained by deletingthe ith row and jth column of the original matrix that is, the rowand column containing aij . Also associated with each element aij isthe cofactor Cij defined by the equation
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
A−1 is the notation for the inverse of A, but keep in mind that
A−1 6= 1
A
There are various ways to compute A−1 from A, assuming that itexists. One way is the cofactor expansion .
Associated with each element aij of a given matrix is the minor Mij
from, which is the determinant of the matrix obtained by deletingthe ith row and jth column of the original matrix that is, the rowand column containing aij . Also associated with each element aij isthe cofactor Cij defined by the equation
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
A−1 is the notation for the inverse of A, but keep in mind that
A−1 6= 1
A
There are various ways to compute A−1 from A, assuming that itexists. One way is the cofactor expansion .
Associated with each element aij of a given matrix
is the minor Mij
from, which is the determinant of the matrix obtained by deletingthe ith row and jth column of the original matrix that is, the rowand column containing aij . Also associated with each element aij isthe cofactor Cij defined by the equation
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
A−1 is the notation for the inverse of A, but keep in mind that
A−1 6= 1
A
There are various ways to compute A−1 from A, assuming that itexists. One way is the cofactor expansion .
Associated with each element aij of a given matrix is the minor Mij
from,
which is the determinant of the matrix obtained by deletingthe ith row and jth column of the original matrix that is, the rowand column containing aij . Also associated with each element aij isthe cofactor Cij defined by the equation
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
A−1 is the notation for the inverse of A, but keep in mind that
A−1 6= 1
A
There are various ways to compute A−1 from A, assuming that itexists. One way is the cofactor expansion .
Associated with each element aij of a given matrix is the minor Mij
from, which is the determinant of the matrix obtained by deletingthe ith row and
jth column of the original matrix that is, the rowand column containing aij . Also associated with each element aij isthe cofactor Cij defined by the equation
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
A−1 is the notation for the inverse of A, but keep in mind that
A−1 6= 1
A
There are various ways to compute A−1 from A, assuming that itexists. One way is the cofactor expansion .
Associated with each element aij of a given matrix is the minor Mij
from, which is the determinant of the matrix obtained by deletingthe ith row and jth column of the original matrix
that is, the rowand column containing aij . Also associated with each element aij isthe cofactor Cij defined by the equation
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
A−1 is the notation for the inverse of A, but keep in mind that
A−1 6= 1
A
There are various ways to compute A−1 from A, assuming that itexists. One way is the cofactor expansion .
Associated with each element aij of a given matrix is the minor Mij
from, which is the determinant of the matrix obtained by deletingthe ith row and jth column of the original matrix that is, the rowand column containing aij .
Also associated with each element aij isthe cofactor Cij defined by the equation
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
OBS
A−1 is the notation for the inverse of A, but keep in mind that
A−1 6= 1
A
There are various ways to compute A−1 from A, assuming that itexists. One way is the cofactor expansion .
Associated with each element aij of a given matrix is the minor Mij
from, which is the determinant of the matrix obtained by deletingthe ith row and jth column of the original matrix that is, the rowand column containing aij . Also associated with each element aij isthe cofactor Cij defined by the equation
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Cij = (−1)nMij
If B = A−1, then it can be shown that the general element bij isgiven by
bij =Cij
det(A)
In general the use of the above equation is not an efficient way tocalculate A−1, instead we can use elementary row operations.There are three such operations:
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Cij = (−1)nMij
If B = A−1, then it can be shown that the general element bij isgiven by
bij =Cij
det(A)
In general the use of the above equation is not an efficient way tocalculate A−1, instead we can use elementary row operations.There are three such operations:
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Cij = (−1)nMij
If B = A−1, then it can be shown that
the general element bij isgiven by
bij =Cij
det(A)
In general the use of the above equation is not an efficient way tocalculate A−1, instead we can use elementary row operations.There are three such operations:
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Cij = (−1)nMij
If B = A−1, then it can be shown that the general element bij
isgiven by
bij =Cij
det(A)
In general the use of the above equation is not an efficient way tocalculate A−1, instead we can use elementary row operations.There are three such operations:
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Cij = (−1)nMij
If B = A−1, then it can be shown that the general element bij isgiven by
bij =Cij
det(A)
In general the use of the above equation is not an efficient way tocalculate A−1, instead we can use elementary row operations.There are three such operations:
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Cij = (−1)nMij
If B = A−1, then it can be shown that the general element bij isgiven by
bij =Cij
det(A)
In general the use of the above equation is not an efficient way tocalculate A−1, instead we can use elementary row operations.There are three such operations:
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Cij = (−1)nMij
If B = A−1, then it can be shown that the general element bij isgiven by
bij =Cij
det(A)
In general the use of the above equation is not an efficient way tocalculate A−1, instead
we can use elementary row operations.There are three such operations:
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Cij = (−1)nMij
If B = A−1, then it can be shown that the general element bij isgiven by
bij =Cij
det(A)
In general the use of the above equation is not an efficient way tocalculate A−1, instead we can use elementary row operations.
There are three such operations:
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Cij = (−1)nMij
If B = A−1, then it can be shown that the general element bij isgiven by
bij =Cij
det(A)
In general the use of the above equation is not an efficient way tocalculate A−1, instead we can use elementary row operations.There are three such operations:
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
1. Interchange of two rows.
2. Multiplication of a row by a nonzero scalar.
3. Addition of any multiple of one row to another row.
The transformation of a matrix by a sequence of elementary rowoperations is referred to as row reduction or Gaussian elimination.Starting with the matrix A we build the m × 2n Augment Matrix
[A | I]
and using elementary row operations we tranforme it into[I | A−1
]
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
1. Interchange of two rows.
2. Multiplication of a row by a nonzero scalar.
3. Addition of any multiple of one row to another row.
The transformation of a matrix by a sequence of elementary rowoperations is referred to as row reduction or Gaussian elimination.Starting with the matrix A we build the m × 2n Augment Matrix
[A | I]
and using elementary row operations we tranforme it into[I | A−1
]
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
1. Interchange of two rows.
2. Multiplication of a row by a nonzero scalar.
3. Addition of any multiple of one row to another row.
The transformation of a matrix by a sequence of elementary rowoperations is referred to as row reduction or Gaussian elimination.Starting with the matrix A we build the m × 2n Augment Matrix
[A | I]
and using elementary row operations we tranforme it into[I | A−1
]
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
1. Interchange of two rows.
2. Multiplication of a row by a nonzero scalar.
3. Addition of any multiple of one row to another row.
The transformation of a matrix by a sequence of elementary rowoperations is referred to as row reduction or Gaussian elimination.Starting with the matrix A we build the m × 2n Augment Matrix
[A | I]
and using elementary row operations we tranforme it into[I | A−1
]
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
1. Interchange of two rows.
2. Multiplication of a row by a nonzero scalar.
3. Addition of any multiple of one row to another row.
The transformation of a matrix by
a sequence of elementary rowoperations is referred to as row reduction or Gaussian elimination.Starting with the matrix A we build the m × 2n Augment Matrix
[A | I]
and using elementary row operations we tranforme it into[I | A−1
]
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
1. Interchange of two rows.
2. Multiplication of a row by a nonzero scalar.
3. Addition of any multiple of one row to another row.
The transformation of a matrix by a sequence of elementary rowoperations
is referred to as row reduction or Gaussian elimination.Starting with the matrix A we build the m × 2n Augment Matrix
[A | I]
and using elementary row operations we tranforme it into[I | A−1
]
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
1. Interchange of two rows.
2. Multiplication of a row by a nonzero scalar.
3. Addition of any multiple of one row to another row.
The transformation of a matrix by a sequence of elementary rowoperations is referred to as row reduction or Gaussian elimination.
Starting with the matrix A we build the m × 2n Augment Matrix
[A | I]
and using elementary row operations we tranforme it into[I | A−1
]
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
1. Interchange of two rows.
2. Multiplication of a row by a nonzero scalar.
3. Addition of any multiple of one row to another row.
The transformation of a matrix by a sequence of elementary rowoperations is referred to as row reduction or Gaussian elimination.Starting with the matrix A
we build the m × 2n Augment Matrix
[A | I]
and using elementary row operations we tranforme it into[I | A−1
]
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
1. Interchange of two rows.
2. Multiplication of a row by a nonzero scalar.
3. Addition of any multiple of one row to another row.
The transformation of a matrix by a sequence of elementary rowoperations is referred to as row reduction or Gaussian elimination.Starting with the matrix A we build the m × 2n Augment Matrix
[A | I]
and using elementary row operations we tranforme it into[I | A−1
]
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
1. Interchange of two rows.
2. Multiplication of a row by a nonzero scalar.
3. Addition of any multiple of one row to another row.
The transformation of a matrix by a sequence of elementary rowoperations is referred to as row reduction or Gaussian elimination.Starting with the matrix A we build the m × 2n Augment Matrix
[A | I]
and using elementary row operations we tranforme it into[I | A−1
]
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
1. Interchange of two rows.
2. Multiplication of a row by a nonzero scalar.
3. Addition of any multiple of one row to another row.
The transformation of a matrix by a sequence of elementary rowoperations is referred to as row reduction or Gaussian elimination.Starting with the matrix A we build the m × 2n Augment Matrix
[A | I]
and using elementary row operations we tranforme it into
[I | A−1
]
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
1. Interchange of two rows.
2. Multiplication of a row by a nonzero scalar.
3. Addition of any multiple of one row to another row.
The transformation of a matrix by a sequence of elementary rowoperations is referred to as row reduction or Gaussian elimination.Starting with the matrix A we build the m × 2n Augment Matrix
[A | I]
and using elementary row operations we tranforme it into[I | A−1
]Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.4
Find the inverse of
A =
1 −1 −13 −1 22 2 3
Solution
First of all, let’s build the augmented matrix
A =
1 −1 −1
∣∣∣ 1 0 0
3 −1 2∣∣∣ 0 1 0
2 2 3∣∣∣ 0 0 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.4
Find the inverse of
A =
1 −1 −13 −1 22 2 3
Solution
First of all, let’s build the augmented matrix
A =
1 −1 −1
∣∣∣ 1 0 0
3 −1 2∣∣∣ 0 1 0
2 2 3∣∣∣ 0 0 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.4
Find the inverse of
A =
1 −1 −13 −1 22 2 3
Solution
First of all, let’s build the augmented matrix
A =
1 −1 −1
∣∣∣ 1 0 0
3 −1 2∣∣∣ 0 1 0
2 2 3∣∣∣ 0 0 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.4
Find the inverse of
A =
1 −1 −13 −1 22 2 3
Solution
First of all, let’s build the augmented matrix
A =
1 −1 −1
∣∣∣ 1 0 0
3 −1 2∣∣∣ 0 1 0
2 2 3∣∣∣ 0 0 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.4
Find the inverse of
A =
1 −1 −13 −1 22 2 3
Solution
First of all, let’s build the augmented matrix
A =
1 −1 −1
∣∣∣ 1 0 0
3 −1 2∣∣∣ 0 1 0
2 2 3∣∣∣ 0 0 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(a) Obtain zeros in the off-diagonal positions in the first column byadding (−3) times the first row to the second row and adding(−2) times the first row to the third row.
A =
1 −1 −1
∣∣∣ 1 0 0
0 2 5∣∣∣ −3 1 0
0 4 5∣∣∣ −2 0 1
(b) Obtain a 1 in the diagonal position in the second column bymultiplying the second row by 1/2 .
A =
1 −1 −1
∣∣∣ 1 0 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 4 5∣∣∣ −2 0 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(a) Obtain zeros in the off-diagonal positions in the first column byadding
(−3) times the first row to the second row and adding(−2) times the first row to the third row.
A =
1 −1 −1
∣∣∣ 1 0 0
0 2 5∣∣∣ −3 1 0
0 4 5∣∣∣ −2 0 1
(b) Obtain a 1 in the diagonal position in the second column bymultiplying the second row by 1/2 .
A =
1 −1 −1
∣∣∣ 1 0 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 4 5∣∣∣ −2 0 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(a) Obtain zeros in the off-diagonal positions in the first column byadding (−3) times the first row
to the second row and adding(−2) times the first row to the third row.
A =
1 −1 −1
∣∣∣ 1 0 0
0 2 5∣∣∣ −3 1 0
0 4 5∣∣∣ −2 0 1
(b) Obtain a 1 in the diagonal position in the second column bymultiplying the second row by 1/2 .
A =
1 −1 −1
∣∣∣ 1 0 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 4 5∣∣∣ −2 0 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(a) Obtain zeros in the off-diagonal positions in the first column byadding (−3) times the first row to the second row and
adding(−2) times the first row to the third row.
A =
1 −1 −1
∣∣∣ 1 0 0
0 2 5∣∣∣ −3 1 0
0 4 5∣∣∣ −2 0 1
(b) Obtain a 1 in the diagonal position in the second column bymultiplying the second row by 1/2 .
A =
1 −1 −1
∣∣∣ 1 0 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 4 5∣∣∣ −2 0 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(a) Obtain zeros in the off-diagonal positions in the first column byadding (−3) times the first row to the second row and adding(−2) times the first row
to the third row.
A =
1 −1 −1
∣∣∣ 1 0 0
0 2 5∣∣∣ −3 1 0
0 4 5∣∣∣ −2 0 1
(b) Obtain a 1 in the diagonal position in the second column bymultiplying the second row by 1/2 .
A =
1 −1 −1
∣∣∣ 1 0 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 4 5∣∣∣ −2 0 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(a) Obtain zeros in the off-diagonal positions in the first column byadding (−3) times the first row to the second row and adding(−2) times the first row to the third row.
A =
1 −1 −1
∣∣∣ 1 0 0
0 2 5∣∣∣ −3 1 0
0 4 5∣∣∣ −2 0 1
(b) Obtain a 1 in the diagonal position in the second column bymultiplying the second row by 1/2 .
A =
1 −1 −1
∣∣∣ 1 0 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 4 5∣∣∣ −2 0 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(a) Obtain zeros in the off-diagonal positions in the first column byadding (−3) times the first row to the second row and adding(−2) times the first row to the third row.
A =
1 −1 −1
∣∣∣ 1 0 0
0 2 5∣∣∣ −3 1 0
0 4 5∣∣∣ −2 0 1
(b) Obtain a 1 in the diagonal position in the second column bymultiplying the second row by 1/2 .
A =
1 −1 −1
∣∣∣ 1 0 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 4 5∣∣∣ −2 0 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(a) Obtain zeros in the off-diagonal positions in the first column byadding (−3) times the first row to the second row and adding(−2) times the first row to the third row.
A =
1 −1 −1
∣∣∣ 1 0 0
0 2 5∣∣∣ −3 1 0
0 4 5∣∣∣ −2 0 1
(b) Obtain a 1 in the diagonal position in the second column
bymultiplying the second row by 1/2 .
A =
1 −1 −1
∣∣∣ 1 0 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 4 5∣∣∣ −2 0 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(a) Obtain zeros in the off-diagonal positions in the first column byadding (−3) times the first row to the second row and adding(−2) times the first row to the third row.
A =
1 −1 −1
∣∣∣ 1 0 0
0 2 5∣∣∣ −3 1 0
0 4 5∣∣∣ −2 0 1
(b) Obtain a 1 in the diagonal position in the second column bymultiplying the second row by 1/2 .
A =
1 −1 −1
∣∣∣ 1 0 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 4 5∣∣∣ −2 0 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(a) Obtain zeros in the off-diagonal positions in the first column byadding (−3) times the first row to the second row and adding(−2) times the first row to the third row.
A =
1 −1 −1
∣∣∣ 1 0 0
0 2 5∣∣∣ −3 1 0
0 4 5∣∣∣ −2 0 1
(b) Obtain a 1 in the diagonal position in the second column bymultiplying the second row by 1/2 .
A =
1 −1 −1
∣∣∣ 1 0 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 4 5∣∣∣ −2 0 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(c) Obtain zeros in the off-diagonal positions in the second columnby adding the second row to the first row and adding (−4) timesthe second row to the third row
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 −5∣∣∣ 4 −2 1
(d) Obtain a 1 in the diagonal position in the third column bymultiplying the third row by (−1/5).
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 1∣∣∣ −4/5 2/5 −1/5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(c) Obtain zeros in the off-diagonal positions in the second columnby adding
the second row to the first row and adding (−4) timesthe second row to the third row
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 −5∣∣∣ 4 −2 1
(d) Obtain a 1 in the diagonal position in the third column bymultiplying the third row by (−1/5).
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 1∣∣∣ −4/5 2/5 −1/5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(c) Obtain zeros in the off-diagonal positions in the second columnby adding the second row
to the first row and adding (−4) timesthe second row to the third row
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 −5∣∣∣ 4 −2 1
(d) Obtain a 1 in the diagonal position in the third column bymultiplying the third row by (−1/5).
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 1∣∣∣ −4/5 2/5 −1/5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(c) Obtain zeros in the off-diagonal positions in the second columnby adding the second row to the first row and
adding (−4) timesthe second row to the third row
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 −5∣∣∣ 4 −2 1
(d) Obtain a 1 in the diagonal position in the third column bymultiplying the third row by (−1/5).
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 1∣∣∣ −4/5 2/5 −1/5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(c) Obtain zeros in the off-diagonal positions in the second columnby adding the second row to the first row and adding (−4) timesthe second row
to the third row
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 −5∣∣∣ 4 −2 1
(d) Obtain a 1 in the diagonal position in the third column bymultiplying the third row by (−1/5).
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 1∣∣∣ −4/5 2/5 −1/5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(c) Obtain zeros in the off-diagonal positions in the second columnby adding the second row to the first row and adding (−4) timesthe second row to the third row
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 −5∣∣∣ 4 −2 1
(d) Obtain a 1 in the diagonal position in the third column bymultiplying the third row by (−1/5).
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 1∣∣∣ −4/5 2/5 −1/5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(c) Obtain zeros in the off-diagonal positions in the second columnby adding the second row to the first row and adding (−4) timesthe second row to the third row
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 −5∣∣∣ 4 −2 1
(d) Obtain a 1 in the diagonal position in the third column bymultiplying the third row by (−1/5).
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 1∣∣∣ −4/5 2/5 −1/5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(c) Obtain zeros in the off-diagonal positions in the second columnby adding the second row to the first row and adding (−4) timesthe second row to the third row
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 −5∣∣∣ 4 −2 1
(d) Obtain a 1 in the diagonal position in the third column
bymultiplying the third row by (−1/5).
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 1∣∣∣ −4/5 2/5 −1/5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(c) Obtain zeros in the off-diagonal positions in the second columnby adding the second row to the first row and adding (−4) timesthe second row to the third row
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 −5∣∣∣ 4 −2 1
(d) Obtain a 1 in the diagonal position in the third column bymultiplying the third row
by (−1/5).
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 1∣∣∣ −4/5 2/5 −1/5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(c) Obtain zeros in the off-diagonal positions in the second columnby adding the second row to the first row and adding (−4) timesthe second row to the third row
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 −5∣∣∣ 4 −2 1
(d) Obtain a 1 in the diagonal position in the third column bymultiplying the third row by (−1/5).
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 1∣∣∣ −4/5 2/5 −1/5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(c) Obtain zeros in the off-diagonal positions in the second columnby adding the second row to the first row and adding (−4) timesthe second row to the third row
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 −5∣∣∣ 4 −2 1
(d) Obtain a 1 in the diagonal position in the third column bymultiplying the third row by (−1/5).
A =
1 0 3/2
∣∣∣ −1/2 1/2 0
0 1 5/2∣∣∣ −3/2 1/2 0
0 0 1∣∣∣ −4/5 2/5 −1/5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(e) Obtain zeros in the off-diagonal positions in the third columnby adding (−3/2) times the third row to the first row and adding(−5/2) times the third row to the second row.
A =
1 0 0
∣∣∣ 7/10 −1/10 3/10
0 1 0∣∣∣ 1/2 −1/2 1/2
0 0 1∣∣∣ −4/5 2/5 −1/5
so, the inverse matrix A−1 is given by
A−1 =
7/10 −1/10 3/101/2 −1/2 1/2−4/5 2/5 −1/5
=1
10
7 −1 35 −5 5−8 4 −2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(e) Obtain zeros in the off-diagonal positions in the third column
by adding (−3/2) times the third row to the first row and adding(−5/2) times the third row to the second row.
A =
1 0 0
∣∣∣ 7/10 −1/10 3/10
0 1 0∣∣∣ 1/2 −1/2 1/2
0 0 1∣∣∣ −4/5 2/5 −1/5
so, the inverse matrix A−1 is given by
A−1 =
7/10 −1/10 3/101/2 −1/2 1/2−4/5 2/5 −1/5
=1
10
7 −1 35 −5 5−8 4 −2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(e) Obtain zeros in the off-diagonal positions in the third columnby adding (−3/2) times the third row
to the first row and adding(−5/2) times the third row to the second row.
A =
1 0 0
∣∣∣ 7/10 −1/10 3/10
0 1 0∣∣∣ 1/2 −1/2 1/2
0 0 1∣∣∣ −4/5 2/5 −1/5
so, the inverse matrix A−1 is given by
A−1 =
7/10 −1/10 3/101/2 −1/2 1/2−4/5 2/5 −1/5
=1
10
7 −1 35 −5 5−8 4 −2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(e) Obtain zeros in the off-diagonal positions in the third columnby adding (−3/2) times the third row to the first row
and adding(−5/2) times the third row to the second row.
A =
1 0 0
∣∣∣ 7/10 −1/10 3/10
0 1 0∣∣∣ 1/2 −1/2 1/2
0 0 1∣∣∣ −4/5 2/5 −1/5
so, the inverse matrix A−1 is given by
A−1 =
7/10 −1/10 3/101/2 −1/2 1/2−4/5 2/5 −1/5
=1
10
7 −1 35 −5 5−8 4 −2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(e) Obtain zeros in the off-diagonal positions in the third columnby adding (−3/2) times the third row to the first row and adding(−5/2) times the third row
to the second row.
A =
1 0 0
∣∣∣ 7/10 −1/10 3/10
0 1 0∣∣∣ 1/2 −1/2 1/2
0 0 1∣∣∣ −4/5 2/5 −1/5
so, the inverse matrix A−1 is given by
A−1 =
7/10 −1/10 3/101/2 −1/2 1/2−4/5 2/5 −1/5
=1
10
7 −1 35 −5 5−8 4 −2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(e) Obtain zeros in the off-diagonal positions in the third columnby adding (−3/2) times the third row to the first row and adding(−5/2) times the third row to the second row.
A =
1 0 0
∣∣∣ 7/10 −1/10 3/10
0 1 0∣∣∣ 1/2 −1/2 1/2
0 0 1∣∣∣ −4/5 2/5 −1/5
so, the inverse matrix A−1 is given by
A−1 =
7/10 −1/10 3/101/2 −1/2 1/2−4/5 2/5 −1/5
=1
10
7 −1 35 −5 5−8 4 −2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(e) Obtain zeros in the off-diagonal positions in the third columnby adding (−3/2) times the third row to the first row and adding(−5/2) times the third row to the second row.
A =
1 0 0
∣∣∣ 7/10 −1/10 3/10
0 1 0∣∣∣ 1/2 −1/2 1/2
0 0 1∣∣∣ −4/5 2/5 −1/5
so, the inverse matrix A−1 is given by
A−1 =
7/10 −1/10 3/101/2 −1/2 1/2−4/5 2/5 −1/5
=1
10
7 −1 35 −5 5−8 4 −2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(e) Obtain zeros in the off-diagonal positions in the third columnby adding (−3/2) times the third row to the first row and adding(−5/2) times the third row to the second row.
A =
1 0 0
∣∣∣ 7/10 −1/10 3/10
0 1 0∣∣∣ 1/2 −1/2 1/2
0 0 1∣∣∣ −4/5 2/5 −1/5
so, the inverse matrix A−1 is given by
A−1 =
7/10 −1/10 3/101/2 −1/2 1/2−4/5 2/5 −1/5
=1
10
7 −1 35 −5 5−8 4 −2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(e) Obtain zeros in the off-diagonal positions in the third columnby adding (−3/2) times the third row to the first row and adding(−5/2) times the third row to the second row.
A =
1 0 0
∣∣∣ 7/10 −1/10 3/10
0 1 0∣∣∣ 1/2 −1/2 1/2
0 0 1∣∣∣ −4/5 2/5 −1/5
so, the inverse matrix A−1 is given by
A−1 =
7/10 −1/10 3/101/2 −1/2 1/2−4/5 2/5 −1/5
=
1
10
7 −1 35 −5 5−8 4 −2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
(e) Obtain zeros in the off-diagonal positions in the third columnby adding (−3/2) times the third row to the first row and adding(−5/2) times the third row to the second row.
A =
1 0 0
∣∣∣ 7/10 −1/10 3/10
0 1 0∣∣∣ 1/2 −1/2 1/2
0 0 1∣∣∣ −4/5 2/5 −1/5
so, the inverse matrix A−1 is given by
A−1 =
7/10 −1/10 3/101/2 −1/2 1/2−4/5 2/5 −1/5
=1
10
7 −1 35 −5 5−8 4 −2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Functions .We sometimes need to consider vectors or matrices whoseelements are functions of a real variable t. In that case, we write
X(t) =
x1(t)x2(t)
...xn(t)
= A(t) =
a11(t) · · · a1n(t)...
...am1(t) amn(t)
respectively.
Continuity
The matrix A(t) is said to be continuous at t = t0 or on aninterval α < t < β if each element of A(t) is a continuous functionat the given point or on the given interval.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Functions .
We sometimes need to consider vectors or matrices whoseelements are functions of a real variable t. In that case, we write
X(t) =
x1(t)x2(t)
...xn(t)
= A(t) =
a11(t) · · · a1n(t)...
...am1(t) amn(t)
respectively.
Continuity
The matrix A(t) is said to be continuous at t = t0 or on aninterval α < t < β if each element of A(t) is a continuous functionat the given point or on the given interval.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Functions .We sometimes need to consider vectors or
matrices whoseelements are functions of a real variable t. In that case, we write
X(t) =
x1(t)x2(t)
...xn(t)
= A(t) =
a11(t) · · · a1n(t)...
...am1(t) amn(t)
respectively.
Continuity
The matrix A(t) is said to be continuous at t = t0 or on aninterval α < t < β if each element of A(t) is a continuous functionat the given point or on the given interval.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Functions .We sometimes need to consider vectors or matrices whoseelements are
functions of a real variable t. In that case, we write
X(t) =
x1(t)x2(t)
...xn(t)
= A(t) =
a11(t) · · · a1n(t)...
...am1(t) amn(t)
respectively.
Continuity
The matrix A(t) is said to be continuous at t = t0 or on aninterval α < t < β if each element of A(t) is a continuous functionat the given point or on the given interval.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Functions .We sometimes need to consider vectors or matrices whoseelements are functions of a real variable t.
In that case, we write
X(t) =
x1(t)x2(t)
...xn(t)
= A(t) =
a11(t) · · · a1n(t)...
...am1(t) amn(t)
respectively.
Continuity
The matrix A(t) is said to be continuous at t = t0 or on aninterval α < t < β if each element of A(t) is a continuous functionat the given point or on the given interval.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Functions .We sometimes need to consider vectors or matrices whoseelements are functions of a real variable t. In that case, we write
X(t) =
x1(t)x2(t)
...xn(t)
= A(t) =
a11(t) · · · a1n(t)...
...am1(t) amn(t)
respectively.
Continuity
The matrix A(t) is said to be continuous at t = t0 or on aninterval α < t < β if each element of A(t) is a continuous functionat the given point or on the given interval.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Functions .We sometimes need to consider vectors or matrices whoseelements are functions of a real variable t. In that case, we write
X(t) =
x1(t)x2(t)
...xn(t)
=
A(t) =
a11(t) · · · a1n(t)...
...am1(t) amn(t)
respectively.
Continuity
The matrix A(t) is said to be continuous at t = t0 or on aninterval α < t < β if each element of A(t) is a continuous functionat the given point or on the given interval.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Functions .We sometimes need to consider vectors or matrices whoseelements are functions of a real variable t. In that case, we write
X(t) =
x1(t)x2(t)
...xn(t)
= A(t) =
a11(t) · · · a1n(t)...
...am1(t) amn(t)
respectively.
Continuity
The matrix A(t) is said to be continuous at t = t0 or on aninterval α < t < β if each element of A(t) is a continuous functionat the given point or on the given interval.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Functions .We sometimes need to consider vectors or matrices whoseelements are functions of a real variable t. In that case, we write
X(t) =
x1(t)x2(t)
...xn(t)
= A(t) =
a11(t) · · · a1n(t)...
...am1(t) amn(t)
respectively.
Continuity
The matrix A(t) is said to be continuous at t = t0 or on aninterval α < t < β if each element of A(t) is a continuous functionat the given point or on the given interval.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Functions .We sometimes need to consider vectors or matrices whoseelements are functions of a real variable t. In that case, we write
X(t) =
x1(t)x2(t)
...xn(t)
= A(t) =
a11(t) · · · a1n(t)...
...am1(t) amn(t)
respectively.
Continuity
The matrix A(t) is said to be continuous at t = t0 or on aninterval α < t < β if each element of A(t) is a continuous functionat the given point or on the given interval.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Functions .We sometimes need to consider vectors or matrices whoseelements are functions of a real variable t. In that case, we write
X(t) =
x1(t)x2(t)
...xn(t)
= A(t) =
a11(t) · · · a1n(t)...
...am1(t) amn(t)
respectively.
Continuity
The matrix A(t) is said to be continuous at t = t0 or
on aninterval α < t < β if each element of A(t) is a continuous functionat the given point or on the given interval.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Functions .We sometimes need to consider vectors or matrices whoseelements are functions of a real variable t. In that case, we write
X(t) =
x1(t)x2(t)
...xn(t)
= A(t) =
a11(t) · · · a1n(t)...
...am1(t) amn(t)
respectively.
Continuity
The matrix A(t) is said to be continuous at t = t0 or on aninterval α < t < β if
each element of A(t) is a continuous functionat the given point or on the given interval.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Functions .We sometimes need to consider vectors or matrices whoseelements are functions of a real variable t. In that case, we write
X(t) =
x1(t)x2(t)
...xn(t)
= A(t) =
a11(t) · · · a1n(t)...
...am1(t) amn(t)
respectively.
Continuity
The matrix A(t) is said to be continuous at t = t0 or on aninterval α < t < β if each element of A(t) is a continuous function
at the given point or on the given interval.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Matrix Functions .We sometimes need to consider vectors or matrices whoseelements are functions of a real variable t. In that case, we write
X(t) =
x1(t)x2(t)
...xn(t)
= A(t) =
a11(t) · · · a1n(t)...
...am1(t) amn(t)
respectively.
Continuity
The matrix A(t) is said to be continuous at t = t0 or on aninterval α < t < β if each element of A(t) is a continuous functionat the given point or on the given interval.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Differentiability
Similarly, A(t) is said to be differentiable if each of its elements isdifferentiable, and its derivative dA(t)/dt is defined by
dA(t)
dt=
(daij(t)
dt
)m×n
that is, each element of dA(t)/dt is the derivative of thecorresponding element of A(t).
Integrability
In the same way, the integral of a matrix function is defined as∫ b
aA(t)dt =
(∫ b
aaij(t)dt
)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Differentiability
Similarly, A(t) is said to be differentiable if each of its elements isdifferentiable, and its derivative dA(t)/dt is defined by
dA(t)
dt=
(daij(t)
dt
)m×n
that is, each element of dA(t)/dt is the derivative of thecorresponding element of A(t).
Integrability
In the same way, the integral of a matrix function is defined as∫ b
aA(t)dt =
(∫ b
aaij(t)dt
)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Differentiability
Similarly, A(t) is said to be differentiable if
each of its elements isdifferentiable, and its derivative dA(t)/dt is defined by
dA(t)
dt=
(daij(t)
dt
)m×n
that is, each element of dA(t)/dt is the derivative of thecorresponding element of A(t).
Integrability
In the same way, the integral of a matrix function is defined as∫ b
aA(t)dt =
(∫ b
aaij(t)dt
)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Differentiability
Similarly, A(t) is said to be differentiable if each of its elements isdifferentiable, and
its derivative dA(t)/dt is defined by
dA(t)
dt=
(daij(t)
dt
)m×n
that is, each element of dA(t)/dt is the derivative of thecorresponding element of A(t).
Integrability
In the same way, the integral of a matrix function is defined as∫ b
aA(t)dt =
(∫ b
aaij(t)dt
)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Differentiability
Similarly, A(t) is said to be differentiable if each of its elements isdifferentiable, and its derivative dA(t)/dt is defined by
dA(t)
dt=
(daij(t)
dt
)m×n
that is, each element of dA(t)/dt is the derivative of thecorresponding element of A(t).
Integrability
In the same way, the integral of a matrix function is defined as∫ b
aA(t)dt =
(∫ b
aaij(t)dt
)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Differentiability
Similarly, A(t) is said to be differentiable if each of its elements isdifferentiable, and its derivative dA(t)/dt is defined by
dA(t)
dt=
(daij(t)
dt
)m×n
that is, each element of dA(t)/dt is the derivative of thecorresponding element of A(t).
Integrability
In the same way, the integral of a matrix function is defined as∫ b
aA(t)dt =
(∫ b
aaij(t)dt
)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Differentiability
Similarly, A(t) is said to be differentiable if each of its elements isdifferentiable, and its derivative dA(t)/dt is defined by
dA(t)
dt=
(daij(t)
dt
)m×n
that is, each element of dA(t)/dt is the derivative of thecorresponding element of A(t).
Integrability
In the same way, the integral of a matrix function is defined as∫ b
aA(t)dt =
(∫ b
aaij(t)dt
)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Differentiability
Similarly, A(t) is said to be differentiable if each of its elements isdifferentiable, and its derivative dA(t)/dt is defined by
dA(t)
dt=
(daij(t)
dt
)m×n
that is, each element of dA(t)/dt
is the derivative of thecorresponding element of A(t).
Integrability
In the same way, the integral of a matrix function is defined as∫ b
aA(t)dt =
(∫ b
aaij(t)dt
)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Differentiability
Similarly, A(t) is said to be differentiable if each of its elements isdifferentiable, and its derivative dA(t)/dt is defined by
dA(t)
dt=
(daij(t)
dt
)m×n
that is, each element of dA(t)/dt is the derivative of thecorresponding element of A(t).
Integrability
In the same way, the integral of a matrix function is defined as∫ b
aA(t)dt =
(∫ b
aaij(t)dt
)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Differentiability
Similarly, A(t) is said to be differentiable if each of its elements isdifferentiable, and its derivative dA(t)/dt is defined by
dA(t)
dt=
(daij(t)
dt
)m×n
that is, each element of dA(t)/dt is the derivative of thecorresponding element of A(t).
Integrability
In the same way, the integral of a matrix function is defined as∫ b
aA(t)dt =
(∫ b
aaij(t)dt
)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Differentiability
Similarly, A(t) is said to be differentiable if each of its elements isdifferentiable, and its derivative dA(t)/dt is defined by
dA(t)
dt=
(daij(t)
dt
)m×n
that is, each element of dA(t)/dt is the derivative of thecorresponding element of A(t).
Integrability
In the same way, the integral of a matrix function is defined as
∫ b
aA(t)dt =
(∫ b
aaij(t)dt
)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Differentiability
Similarly, A(t) is said to be differentiable if each of its elements isdifferentiable, and its derivative dA(t)/dt is defined by
dA(t)
dt=
(daij(t)
dt
)m×n
that is, each element of dA(t)/dt is the derivative of thecorresponding element of A(t).
Integrability
In the same way, the integral of a matrix function is defined as∫ b
aA(t)dt =
(∫ b
aaij(t)dt
)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Differentiability
Similarly, A(t) is said to be differentiable if each of its elements isdifferentiable, and its derivative dA(t)/dt is defined by
dA(t)
dt=
(daij(t)
dt
)m×n
that is, each element of dA(t)/dt is the derivative of thecorresponding element of A(t).
Integrability
In the same way, the integral of a matrix function is defined as∫ b
aA(t)dt =
(∫ b
aaij(t)dt
)m×n
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.5
Consider the matrix
A(t) =
(sin(t) 1t cos(t)
)Find A′(t) and
∫ π0 A(t)dt.
Solution
A′(t) =
(cos(t) 0
1 −sin(t)
)∫ π
0A(t)dt =
(∫ π0 sin(t)dt
∫ π0 1dt∫ π
0 tdt∫ π0 cos(t)dt
)=
(2 π
π2/2 0
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.5
Consider the matrix
A(t) =
(sin(t) 1t cos(t)
)Find A′(t) and
∫ π0 A(t)dt.
Solution
A′(t) =
(cos(t) 0
1 −sin(t)
)∫ π
0A(t)dt =
(∫ π0 sin(t)dt
∫ π0 1dt∫ π
0 tdt∫ π0 cos(t)dt
)=
(2 π
π2/2 0
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.5
Consider the matrix
A(t) =
(sin(t) 1t cos(t)
)Find A′(t) and
∫ π0 A(t)dt.
Solution
A′(t) =
(cos(t) 0
1 −sin(t)
)∫ π
0A(t)dt =
(∫ π0 sin(t)dt
∫ π0 1dt∫ π
0 tdt∫ π0 cos(t)dt
)=
(2 π
π2/2 0
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.5
Consider the matrix
A(t) =
(sin(t) 1t cos(t)
)
Find A′(t) and∫ π0 A(t)dt.
Solution
A′(t) =
(cos(t) 0
1 −sin(t)
)∫ π
0A(t)dt =
(∫ π0 sin(t)dt
∫ π0 1dt∫ π
0 tdt∫ π0 cos(t)dt
)=
(2 π
π2/2 0
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.5
Consider the matrix
A(t) =
(sin(t) 1t cos(t)
)Find A′(t) and
∫ π0 A(t)dt.
Solution
A′(t) =
(cos(t) 0
1 −sin(t)
)∫ π
0A(t)dt =
(∫ π0 sin(t)dt
∫ π0 1dt∫ π
0 tdt∫ π0 cos(t)dt
)=
(2 π
π2/2 0
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.5
Consider the matrix
A(t) =
(sin(t) 1t cos(t)
)Find A′(t) and
∫ π0 A(t)dt.
Solution
A′(t) =
(cos(t) 0
1 −sin(t)
)∫ π
0A(t)dt =
(∫ π0 sin(t)dt
∫ π0 1dt∫ π
0 tdt∫ π0 cos(t)dt
)=
(2 π
π2/2 0
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.5
Consider the matrix
A(t) =
(sin(t) 1t cos(t)
)Find A′(t) and
∫ π0 A(t)dt.
Solution
A′(t) =
(cos(t) 0
1 −sin(t)
)
∫ π
0A(t)dt =
(∫ π0 sin(t)dt
∫ π0 1dt∫ π
0 tdt∫ π0 cos(t)dt
)=
(2 π
π2/2 0
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.5
Consider the matrix
A(t) =
(sin(t) 1t cos(t)
)Find A′(t) and
∫ π0 A(t)dt.
Solution
A′(t) =
(cos(t) 0
1 −sin(t)
)∫ π
0A(t)dt =
(∫ π0 sin(t)dt
∫ π0 1dt∫ π
0 tdt∫ π0 cos(t)dt
)=
(2 π
π2/2 0
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.5
Consider the matrix
A(t) =
(sin(t) 1t cos(t)
)Find A′(t) and
∫ π0 A(t)dt.
Solution
A′(t) =
(cos(t) 0
1 −sin(t)
)∫ π
0A(t)dt =
(∫ π0 sin(t)dt
∫ π0 1dt∫ π
0 tdt∫ π0 cos(t)dt
)=
(2 π
π2/2 0
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Review of Matrices
Example 7.5
Consider the matrix
A(t) =
(sin(t) 1t cos(t)
)Find A′(t) and
∫ π0 A(t)dt.
Solution
A′(t) =
(cos(t) 0
1 −sin(t)
)∫ π
0A(t)dt =
(∫ π0 sin(t)dt
∫ π0 1dt∫ π
0 tdt∫ π0 cos(t)dt
)=
(2 π
π2/2 0
)Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Systems of Linear Algebraic Equations . A set of n simultaneouslinear algebraic equations in n variables
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
an1x1 + an2x2 + . . .+ annxn = bnn
can be written as
AX = b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Systems of Linear Algebraic Equations .
A set of n simultaneouslinear algebraic equations in n variables
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
an1x1 + an2x2 + . . .+ annxn = bnn
can be written as
AX = b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Systems of Linear Algebraic Equations . A set of n simultaneouslinear algebraic equations in n variables
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
an1x1 + an2x2 + . . .+ annxn = bnn
can be written as
AX = b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Systems of Linear Algebraic Equations . A set of n simultaneouslinear algebraic equations in n variables
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
an1x1 + an2x2 + . . .+ annxn = bnn
can be written as
AX = b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Systems of Linear Algebraic Equations . A set of n simultaneouslinear algebraic equations in n variables
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
an1x1 + an2x2 + . . .+ annxn = bnn
can be written as
AX = b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Systems of Linear Algebraic Equations . A set of n simultaneouslinear algebraic equations in n variables
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
an1x1 + an2x2 + . . .+ annxn = bnn
can be written as
AX = b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
If b = 0, the system is said to be homogeneous; otherwise, it isnonhomogeneous.
If the matrix A is invertible,hence A−1 exists, and therefore wehave
X = A−1b
In particular, the homogeneous problem AX = b, corresponding tob = 0, has only the trivial solution 0.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
If b = 0, the system is said to be homogeneous;
otherwise, it isnonhomogeneous.
If the matrix A is invertible,hence A−1 exists, and therefore wehave
X = A−1b
In particular, the homogeneous problem AX = b, corresponding tob = 0, has only the trivial solution 0.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
If b = 0, the system is said to be homogeneous; otherwise, it isnonhomogeneous.
If the matrix A is invertible,hence A−1 exists, and therefore wehave
X = A−1b
In particular, the homogeneous problem AX = b, corresponding tob = 0, has only the trivial solution 0.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
If b = 0, the system is said to be homogeneous; otherwise, it isnonhomogeneous.
If the matrix A is invertible,
hence A−1 exists, and therefore wehave
X = A−1b
In particular, the homogeneous problem AX = b, corresponding tob = 0, has only the trivial solution 0.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
If b = 0, the system is said to be homogeneous; otherwise, it isnonhomogeneous.
If the matrix A is invertible,hence A−1 exists, and
therefore wehave
X = A−1b
In particular, the homogeneous problem AX = b, corresponding tob = 0, has only the trivial solution 0.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
If b = 0, the system is said to be homogeneous; otherwise, it isnonhomogeneous.
If the matrix A is invertible,hence A−1 exists, and therefore wehave
X = A−1b
In particular, the homogeneous problem AX = b, corresponding tob = 0, has only the trivial solution 0.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
If b = 0, the system is said to be homogeneous; otherwise, it isnonhomogeneous.
If the matrix A is invertible,hence A−1 exists, and therefore wehave
X = A−1b
In particular, the homogeneous problem AX = b, corresponding tob = 0, has only the trivial solution 0.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
If b = 0, the system is said to be homogeneous; otherwise, it isnonhomogeneous.
If the matrix A is invertible,hence A−1 exists, and therefore wehave
X = A−1b
In particular,
the homogeneous problem AX = b, corresponding tob = 0, has only the trivial solution 0.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
If b = 0, the system is said to be homogeneous; otherwise, it isnonhomogeneous.
If the matrix A is invertible,hence A−1 exists, and therefore wehave
X = A−1b
In particular, the homogeneous problem AX = b,
corresponding tob = 0, has only the trivial solution 0.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
If b = 0, the system is said to be homogeneous; otherwise, it isnonhomogeneous.
If the matrix A is invertible,hence A−1 exists, and therefore wehave
X = A−1b
In particular, the homogeneous problem AX = b, corresponding tob = 0,
has only the trivial solution 0.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
If b = 0, the system is said to be homogeneous; otherwise, it isnonhomogeneous.
If the matrix A is invertible,hence A−1 exists, and therefore wehave
X = A−1b
In particular, the homogeneous problem AX = b, corresponding tob = 0, has only the trivial solution 0.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
On the other hand, if A is singular, A−1 does not exist, so thehomogeneous system
AX = 0
has (infinitely many) nonzero solutions in addition to the trivialsolution.
Solving a Linear System
For solving particular systems, we can form the augmented matrix
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
On the other hand,
if A is singular, A−1 does not exist, so thehomogeneous system
AX = 0
has (infinitely many) nonzero solutions in addition to the trivialsolution.
Solving a Linear System
For solving particular systems, we can form the augmented matrix
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
On the other hand, if A is singular,
A−1 does not exist, so thehomogeneous system
AX = 0
has (infinitely many) nonzero solutions in addition to the trivialsolution.
Solving a Linear System
For solving particular systems, we can form the augmented matrix
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
On the other hand, if A is singular, A−1 does not exist,
so thehomogeneous system
AX = 0
has (infinitely many) nonzero solutions in addition to the trivialsolution.
Solving a Linear System
For solving particular systems, we can form the augmented matrix
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
On the other hand, if A is singular, A−1 does not exist, so thehomogeneous system
AX = 0
has (infinitely many) nonzero solutions in addition to the trivialsolution.
Solving a Linear System
For solving particular systems, we can form the augmented matrix
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
On the other hand, if A is singular, A−1 does not exist, so thehomogeneous system
AX = 0
has (infinitely many) nonzero solutions in addition to the trivialsolution.
Solving a Linear System
For solving particular systems, we can form the augmented matrix
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
On the other hand, if A is singular, A−1 does not exist, so thehomogeneous system
AX = 0
has (infinitely many) nonzero solutions in addition to the trivialsolution.
Solving a Linear System
For solving particular systems, we can form the augmented matrix
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
On the other hand, if A is singular, A−1 does not exist, so thehomogeneous system
AX = 0
has (infinitely many) nonzero solutions in addition to the trivialsolution.
Solving a Linear System
For solving particular systems, we can form the augmented matrix
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
On the other hand, if A is singular, A−1 does not exist, so thehomogeneous system
AX = 0
has (infinitely many) nonzero solutions in addition to the trivialsolution.
Solving a Linear System
For solving particular systems,
we can form the augmented matrix
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
On the other hand, if A is singular, A−1 does not exist, so thehomogeneous system
AX = 0
has (infinitely many) nonzero solutions in addition to the trivialsolution.
Solving a Linear System
For solving particular systems, we can form the augmented matrix
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
[A|b] =
a11 a12 . . . a1n
∣∣∣ b1
a21 a22 . . . a2n
∣∣∣ b2...
∣∣∣ ...
an1 an2 . . . ann
∣∣∣ bn
We now perform row operations on the augmented matrix so as totransform A into an upper triangular matrix.
[U|b̄]
Once this is done, it is easy to see whether the system hassolutions, and to find them if it does.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
[A|b] =
a11 a12 . . . a1n
∣∣∣ b1
a21 a22 . . . a2n
∣∣∣ b2...
∣∣∣ ...
an1 an2 . . . ann
∣∣∣ bn
We now perform row operations on the augmented matrix so as totransform A into an upper triangular matrix.
[U|b̄]
Once this is done, it is easy to see whether the system hassolutions, and to find them if it does.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
[A|b] =
a11 a12 . . . a1n
∣∣∣ b1
a21 a22 . . . a2n
∣∣∣ b2...
∣∣∣ ...
an1 an2 . . . ann
∣∣∣ bn
We now perform row operations on the augmented matrix so as totransform A into an upper triangular matrix.
[U|b̄]
Once this is done, it is easy to see whether the system hassolutions, and to find them if it does.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
[A|b] =
a11 a12 . . . a1n
∣∣∣ b1
a21 a22 . . . a2n
∣∣∣ b2...
∣∣∣ ...
an1 an2 . . . ann
∣∣∣ bn
We now perform row operations on the augmented matrix
so as totransform A into an upper triangular matrix.
[U|b̄]
Once this is done, it is easy to see whether the system hassolutions, and to find them if it does.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
[A|b] =
a11 a12 . . . a1n
∣∣∣ b1
a21 a22 . . . a2n
∣∣∣ b2...
∣∣∣ ...
an1 an2 . . . ann
∣∣∣ bn
We now perform row operations on the augmented matrix so as totransform A
into an upper triangular matrix.
[U|b̄]
Once this is done, it is easy to see whether the system hassolutions, and to find them if it does.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
[A|b] =
a11 a12 . . . a1n
∣∣∣ b1
a21 a22 . . . a2n
∣∣∣ b2...
∣∣∣ ...
an1 an2 . . . ann
∣∣∣ bn
We now perform row operations on the augmented matrix so as totransform A into an upper triangular matrix.
[U|b̄]
Once this is done, it is easy to see whether the system hassolutions, and to find them if it does.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
[A|b] =
a11 a12 . . . a1n
∣∣∣ b1
a21 a22 . . . a2n
∣∣∣ b2...
∣∣∣ ...
an1 an2 . . . ann
∣∣∣ bn
We now perform row operations on the augmented matrix so as totransform A into an upper triangular matrix.
[U|b̄]
Once this is done, it is easy to see whether the system hassolutions, and to find them if it does.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
[A|b] =
a11 a12 . . . a1n
∣∣∣ b1
a21 a22 . . . a2n
∣∣∣ b2...
∣∣∣ ...
an1 an2 . . . ann
∣∣∣ bn
We now perform row operations on the augmented matrix so as totransform A into an upper triangular matrix.
[U|b̄]
Once this is done,
it is easy to see whether the system hassolutions, and to find them if it does.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
[A|b] =
a11 a12 . . . a1n
∣∣∣ b1
a21 a22 . . . a2n
∣∣∣ b2...
∣∣∣ ...
an1 an2 . . . ann
∣∣∣ bn
We now perform row operations on the augmented matrix so as totransform A into an upper triangular matrix.
[U|b̄]
Once this is done, it is easy to see whether the system hassolutions, and
to find them if it does.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
[A|b] =
a11 a12 . . . a1n
∣∣∣ b1
a21 a22 . . . a2n
∣∣∣ b2...
∣∣∣ ...
an1 an2 . . . ann
∣∣∣ bn
We now perform row operations on the augmented matrix so as totransform A into an upper triangular matrix.
[U|b̄]
Once this is done, it is easy to see whether the system hassolutions, and to find them if it does.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.6
Solve the system of equations
x1 − 2x2 + 3x3 = 7−x1 + x2 − 2x3 = −52x1 − x2 − x3 = 4
Solution
The augmented matrix for the system is1 −2 3
∣∣∣ 7
−1 1 −2∣∣∣ −5
2 −1 −1∣∣∣ 4
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.6
Solve the system of equations
x1 − 2x2 + 3x3 = 7−x1 + x2 − 2x3 = −52x1 − x2 − x3 = 4
Solution
The augmented matrix for the system is1 −2 3
∣∣∣ 7
−1 1 −2∣∣∣ −5
2 −1 −1∣∣∣ 4
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.6
Solve the system of equations
x1 − 2x2 + 3x3 = 7−x1 + x2 − 2x3 = −52x1 − x2 − x3 = 4
Solution
The augmented matrix for the system is1 −2 3
∣∣∣ 7
−1 1 −2∣∣∣ −5
2 −1 −1∣∣∣ 4
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.6
Solve the system of equations
x1 − 2x2 + 3x3 = 7−x1 + x2 − 2x3 = −52x1 − x2 − x3 = 4
Solution
The augmented matrix for the system is1 −2 3
∣∣∣ 7
−1 1 −2∣∣∣ −5
2 −1 −1∣∣∣ 4
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.6
Solve the system of equations
x1 − 2x2 + 3x3 = 7−x1 + x2 − 2x3 = −52x1 − x2 − x3 = 4
Solution
The augmented matrix for the system is1 −2 3
∣∣∣ 7
−1 1 −2∣∣∣ −5
2 −1 −1∣∣∣ 4
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.6
Solve the system of equations
x1 − 2x2 + 3x3 = 7−x1 + x2 − 2x3 = −52x1 − x2 − x3 = 4
Solution
The augmented matrix for the system is
1 −2 3
∣∣∣ 7
−1 1 −2∣∣∣ −5
2 −1 −1∣∣∣ 4
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.6
Solve the system of equations
x1 − 2x2 + 3x3 = 7−x1 + x2 − 2x3 = −52x1 − x2 − x3 = 4
Solution
The augmented matrix for the system is1 −2 3
∣∣∣ 7
−1 1 −2∣∣∣ −5
2 −1 −1∣∣∣ 4
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.6
Solve the system of equations
x1 − 2x2 + 3x3 = 7−x1 + x2 − 2x3 = −52x1 − x2 − x3 = 4
Solution
The augmented matrix for the system is1 −2 3
∣∣∣ 7
−1 1 −2∣∣∣ −5
2 −1 −1∣∣∣ 4
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
We now perform row operations on the augmented matrix with aview to introducing zeros in the lower left part of the matrix.
(a) Add the first row to the second row, and add (−2) times thefirst row to the third row.
1 −2 3
∣∣∣ 7
0 −1 1∣∣∣ 2
0 −3 −7∣∣∣ −10
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
We now perform row operations on the augmented matrix
with aview to introducing zeros in the lower left part of the matrix.
(a) Add the first row to the second row, and add (−2) times thefirst row to the third row.
1 −2 3
∣∣∣ 7
0 −1 1∣∣∣ 2
0 −3 −7∣∣∣ −10
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
We now perform row operations on the augmented matrix with aview to introducing zeros in the lower left part of the matrix.
(a) Add the first row to the second row, and add (−2) times thefirst row to the third row.
1 −2 3
∣∣∣ 7
0 −1 1∣∣∣ 2
0 −3 −7∣∣∣ −10
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
We now perform row operations on the augmented matrix with aview to introducing zeros in the lower left part of the matrix.
(a) Add the first row to the second row,
and add (−2) times thefirst row to the third row.
1 −2 3
∣∣∣ 7
0 −1 1∣∣∣ 2
0 −3 −7∣∣∣ −10
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
We now perform row operations on the augmented matrix with aview to introducing zeros in the lower left part of the matrix.
(a) Add the first row to the second row, and add (−2) times thefirst row
to the third row.
1 −2 3
∣∣∣ 7
0 −1 1∣∣∣ 2
0 −3 −7∣∣∣ −10
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
We now perform row operations on the augmented matrix with aview to introducing zeros in the lower left part of the matrix.
(a) Add the first row to the second row, and add (−2) times thefirst row to the third row.
1 −2 3
∣∣∣ 7
0 −1 1∣∣∣ 2
0 −3 −7∣∣∣ −10
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
We now perform row operations on the augmented matrix with aview to introducing zeros in the lower left part of the matrix.
(a) Add the first row to the second row, and add (−2) times thefirst row to the third row.
1 −2 3
∣∣∣ 7
0 −1 1∣∣∣ 2
0 −3 −7∣∣∣ −10
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(b) Multiply the second row by −1.1 −2 3
∣∣∣ 7
0 1 −1∣∣∣ −2
0 3 −7∣∣∣ −10
(c) Add (−3) times the second row to the third row.
1 −2 3∣∣∣ 7
0 1 −1∣∣∣ −2
0 0 −4∣∣∣ −4
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(b) Multiply the second row by −1.
1 −2 3
∣∣∣ 7
0 1 −1∣∣∣ −2
0 3 −7∣∣∣ −10
(c) Add (−3) times the second row to the third row.
1 −2 3∣∣∣ 7
0 1 −1∣∣∣ −2
0 0 −4∣∣∣ −4
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(b) Multiply the second row by −1.1 −2 3
∣∣∣ 7
0 1 −1∣∣∣ −2
0 3 −7∣∣∣ −10
(c) Add (−3) times the second row to the third row.1 −2 3
∣∣∣ 7
0 1 −1∣∣∣ −2
0 0 −4∣∣∣ −4
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(b) Multiply the second row by −1.1 −2 3
∣∣∣ 7
0 1 −1∣∣∣ −2
0 3 −7∣∣∣ −10
(c) Add (−3) times the second row
to the third row.1 −2 3
∣∣∣ 7
0 1 −1∣∣∣ −2
0 0 −4∣∣∣ −4
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(b) Multiply the second row by −1.1 −2 3
∣∣∣ 7
0 1 −1∣∣∣ −2
0 3 −7∣∣∣ −10
(c) Add (−3) times the second row to the third row.
1 −2 3
∣∣∣ 7
0 1 −1∣∣∣ −2
0 0 −4∣∣∣ −4
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(b) Multiply the second row by −1.1 −2 3
∣∣∣ 7
0 1 −1∣∣∣ −2
0 3 −7∣∣∣ −10
(c) Add (−3) times the second row to the third row.
1 −2 3∣∣∣ 7
0 1 −1∣∣∣ −2
0 0 −4∣∣∣ −4
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(d) Divide the third row by −4.1 −2 3
∣∣∣ 7
0 1 −1∣∣∣ −2
0 0 1∣∣∣ 1
The matrix obtained in this manner corresponds to the system ofequations
x1 − 2x2 + 3x3 = 7x2 − x3 = −2
x3 = 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(d) Divide the third row by −4.
1 −2 3
∣∣∣ 7
0 1 −1∣∣∣ −2
0 0 1∣∣∣ 1
The matrix obtained in this manner corresponds to the system ofequations
x1 − 2x2 + 3x3 = 7x2 − x3 = −2
x3 = 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(d) Divide the third row by −4.1 −2 3
∣∣∣ 7
0 1 −1∣∣∣ −2
0 0 1∣∣∣ 1
The matrix obtained in this manner corresponds to the system ofequations
x1 − 2x2 + 3x3 = 7x2 − x3 = −2
x3 = 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(d) Divide the third row by −4.1 −2 3
∣∣∣ 7
0 1 −1∣∣∣ −2
0 0 1∣∣∣ 1
The matrix obtained in this manner corresponds to the system ofequations
x1 − 2x2 + 3x3 = 7x2 − x3 = −2
x3 = 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(d) Divide the third row by −4.1 −2 3
∣∣∣ 7
0 1 −1∣∣∣ −2
0 0 1∣∣∣ 1
The matrix obtained in this manner corresponds to the system ofequations
x1 − 2x2 + 3x3 = 7x2 − x3 = −2
x3 = 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
From the last of equations we have
x3 = 2, x2 = −2 + x3 = − 1, x3 = 7 + 2x2 − 2x3 = 2
Thus, we obtain
X =
2− 1
1
Now, since the solution is unique, we conclude that the coefficientmatrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
From the last of equations we have
x3 = 2, x2 = −2 + x3 = − 1, x3 = 7 + 2x2 − 2x3 = 2
Thus, we obtain
X =
2− 1
1
Now, since the solution is unique, we conclude that the coefficientmatrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
From the last of equations we have
x3 = 2,
x2 = −2 + x3 = − 1, x3 = 7 + 2x2 − 2x3 = 2
Thus, we obtain
X =
2− 1
1
Now, since the solution is unique, we conclude that the coefficientmatrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
From the last of equations we have
x3 = 2, x2 = −2 + x3 =
− 1, x3 = 7 + 2x2 − 2x3 = 2
Thus, we obtain
X =
2− 1
1
Now, since the solution is unique, we conclude that the coefficientmatrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
From the last of equations we have
x3 = 2, x2 = −2 + x3 = − 1,
x3 = 7 + 2x2 − 2x3 = 2
Thus, we obtain
X =
2− 1
1
Now, since the solution is unique, we conclude that the coefficientmatrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
From the last of equations we have
x3 = 2, x2 = −2 + x3 = − 1, x3 = 7 + 2x2 − 2x3 =
2
Thus, we obtain
X =
2− 1
1
Now, since the solution is unique, we conclude that the coefficientmatrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
From the last of equations we have
x3 = 2, x2 = −2 + x3 = − 1, x3 = 7 + 2x2 − 2x3 = 2
Thus, we obtain
X =
2− 1
1
Now, since the solution is unique, we conclude that the coefficientmatrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
From the last of equations we have
x3 = 2, x2 = −2 + x3 = − 1, x3 = 7 + 2x2 − 2x3 = 2
Thus, we obtain
X =
2− 1
1
Now, since the solution is unique, we conclude that the coefficientmatrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
From the last of equations we have
x3 = 2, x2 = −2 + x3 = − 1, x3 = 7 + 2x2 − 2x3 = 2
Thus, we obtain
X =
2− 1
1
Now, since the solution is unique, we conclude that the coefficientmatrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
From the last of equations we have
x3 = 2, x2 = −2 + x3 = − 1, x3 = 7 + 2x2 − 2x3 = 2
Thus, we obtain
X =
2− 1
1
Now, since the solution is unique,
we conclude that the coefficientmatrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
From the last of equations we have
x3 = 2, x2 = −2 + x3 = − 1, x3 = 7 + 2x2 − 2x3 = 2
Thus, we obtain
X =
2− 1
1
Now, since the solution is unique, we conclude that the coefficientmatrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.7
Solve the system of equations
x1 − 2x2 + 3x3 = b1−x1 + x2 − 2x3 = b22x1 − x2 − 3x3 = b3
for various values of b1, b2, and b3
Solution
By performing steps (a), (b), and (c) as in Example 7.6, wetransform the matrix into
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.7
Solve the system of equations
x1 − 2x2 + 3x3 = b1−x1 + x2 − 2x3 = b22x1 − x2 − 3x3 = b3
for various values of b1, b2, and b3
Solution
By performing steps (a), (b), and (c) as in Example 7.6, wetransform the matrix into
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.7
Solve the system of equations
x1 − 2x2 + 3x3 = b1−x1 + x2 − 2x3 = b22x1 − x2 − 3x3 = b3
for various values of b1, b2, and b3
Solution
By performing steps (a), (b), and (c) as in Example 7.6, wetransform the matrix into
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.7
Solve the system of equations
x1 − 2x2 + 3x3 = b1−x1 + x2 − 2x3 = b22x1 − x2 − 3x3 = b3
for various values of b1, b2, and b3
Solution
By performing steps (a), (b), and (c) as in Example 7.6, wetransform the matrix into
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.7
Solve the system of equations
x1 − 2x2 + 3x3 = b1−x1 + x2 − 2x3 = b22x1 − x2 − 3x3 = b3
for various values of b1, b2, and b3
Solution
By performing steps (a), (b), and (c) as in Example 7.6, wetransform the matrix into
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.7
Solve the system of equations
x1 − 2x2 + 3x3 = b1−x1 + x2 − 2x3 = b22x1 − x2 − 3x3 = b3
for various values of b1, b2, and b3
Solution
By performing steps (a), (b), and (c) as in Example 7.6,
wetransform the matrix into
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.7
Solve the system of equations
x1 − 2x2 + 3x3 = b1−x1 + x2 − 2x3 = b22x1 − x2 − 3x3 = b3
for various values of b1, b2, and b3
Solution
By performing steps (a), (b), and (c) as in Example 7.6, wetransform the matrix into
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −2 3
∣∣∣ b1
0 1 −1∣∣∣ −b1 − b2
0 0 0∣∣∣ b1 + 3b2 + b3
The equation corresponding to the third row is
b1 + 3b2 + b3 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −2 3
∣∣∣ b1
0 1 −1∣∣∣ −b1 − b2
0 0 0∣∣∣ b1 + 3b2 + b3
The equation corresponding to the third row is
b1 + 3b2 + b3 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −2 3
∣∣∣ b1
0 1 −1∣∣∣ −b1 − b2
0 0 0∣∣∣ b1 + 3b2 + b3
The equation corresponding to the third row is
b1 + 3b2 + b3 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −2 3
∣∣∣ b1
0 1 −1∣∣∣ −b1 − b2
0 0 0∣∣∣ b1 + 3b2 + b3
The equation corresponding to the third row is
b1 + 3b2 + b3 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
thus the system has no solution unless the above condition issatisfied by b1, b2, and b3.
b1 = −3b2 − b3
Assuming that the condition is satisfied1 −2 3
∣∣∣ −3b2 − b3
0 1 −1∣∣∣ −(−3b2 − b3)− b2
0 0 0∣∣∣ (−3b2 − b3) + 3b2 + b3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
thus the system has no solution unless
the above condition issatisfied by b1, b2, and b3.
b1 = −3b2 − b3
Assuming that the condition is satisfied1 −2 3
∣∣∣ −3b2 − b3
0 1 −1∣∣∣ −(−3b2 − b3)− b2
0 0 0∣∣∣ (−3b2 − b3) + 3b2 + b3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
thus the system has no solution unless the above condition issatisfied by b1, b2, and b3.
b1 = −3b2 − b3
Assuming that the condition is satisfied1 −2 3
∣∣∣ −3b2 − b3
0 1 −1∣∣∣ −(−3b2 − b3)− b2
0 0 0∣∣∣ (−3b2 − b3) + 3b2 + b3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
thus the system has no solution unless the above condition issatisfied by b1, b2, and b3.
b1 = −3b2 − b3
Assuming that the condition is satisfied1 −2 3
∣∣∣ −3b2 − b3
0 1 −1∣∣∣ −(−3b2 − b3)− b2
0 0 0∣∣∣ (−3b2 − b3) + 3b2 + b3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
thus the system has no solution unless the above condition issatisfied by b1, b2, and b3.
b1 = −3b2 − b3
Assuming that the condition is satisfied
1 −2 3
∣∣∣ −3b2 − b3
0 1 −1∣∣∣ −(−3b2 − b3)− b2
0 0 0∣∣∣ (−3b2 − b3) + 3b2 + b3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
thus the system has no solution unless the above condition issatisfied by b1, b2, and b3.
b1 = −3b2 − b3
Assuming that the condition is satisfied1 −2 3
∣∣∣ −3b2 − b3
0 1 −1∣∣∣ −(−3b2 − b3)− b2
0 0 0∣∣∣ (−3b2 − b3) + 3b2 + b3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −2 3
∣∣∣ −3b2 − b3
0 1 −1∣∣∣ 2b2 + b3
0 0 0∣∣∣ 0
Add (2) times the second row to the first row.
1 0 1∣∣∣ −3b2 − b3
0 1 −1∣∣∣ b2 + b3
0 0 0∣∣∣ 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −2 3
∣∣∣ −3b2 − b3
0 1 −1∣∣∣ 2b2 + b3
0 0 0∣∣∣ 0
Add (2) times the second row to the first row.1 0 1
∣∣∣ −3b2 − b3
0 1 −1∣∣∣ b2 + b3
0 0 0∣∣∣ 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −2 3
∣∣∣ −3b2 − b3
0 1 −1∣∣∣ 2b2 + b3
0 0 0∣∣∣ 0
Add (2) times the second row
to the first row.1 0 1
∣∣∣ −3b2 − b3
0 1 −1∣∣∣ b2 + b3
0 0 0∣∣∣ 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −2 3
∣∣∣ −3b2 − b3
0 1 −1∣∣∣ 2b2 + b3
0 0 0∣∣∣ 0
Add (2) times the second row to the first row.
1 0 1
∣∣∣ −3b2 − b3
0 1 −1∣∣∣ b2 + b3
0 0 0∣∣∣ 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −2 3
∣∣∣ −3b2 − b3
0 1 −1∣∣∣ 2b2 + b3
0 0 0∣∣∣ 0
Add (2) times the second row to the first row.
1 0 1∣∣∣ −3b2 − b3
0 1 −1∣∣∣ b2 + b3
0 0 0∣∣∣ 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Thus, we have two equations and one unknown,so one of thevariables let’s say x3, is equal to a parameter α, obtaining thesystem
x1 + α = −3b2 − b3x2 − α = b2 + b3
Hence, we obtain
x1 = −α− 3b2 − b3; x2 = α + b2 + b3
X =
−α− 3b2 − b3α + b2 + b3
α
= α
−111
+
−3b2 − b3b2 + b3
0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Thus, we have two equations and one unknown,
so one of thevariables let’s say x3, is equal to a parameter α, obtaining thesystem
x1 + α = −3b2 − b3x2 − α = b2 + b3
Hence, we obtain
x1 = −α− 3b2 − b3; x2 = α + b2 + b3
X =
−α− 3b2 − b3α + b2 + b3
α
= α
−111
+
−3b2 − b3b2 + b3
0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Thus, we have two equations and one unknown,so one of thevariables let’s say x3,
is equal to a parameter α, obtaining thesystem
x1 + α = −3b2 − b3x2 − α = b2 + b3
Hence, we obtain
x1 = −α− 3b2 − b3; x2 = α + b2 + b3
X =
−α− 3b2 − b3α + b2 + b3
α
= α
−111
+
−3b2 − b3b2 + b3
0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Thus, we have two equations and one unknown,so one of thevariables let’s say x3, is equal to a parameter α, obtaining thesystem
x1 + α = −3b2 − b3x2 − α = b2 + b3
Hence, we obtain
x1 = −α− 3b2 − b3; x2 = α + b2 + b3
X =
−α− 3b2 − b3α + b2 + b3
α
= α
−111
+
−3b2 − b3b2 + b3
0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Thus, we have two equations and one unknown,so one of thevariables let’s say x3, is equal to a parameter α, obtaining thesystem
x1 + α = −3b2 − b3x2 − α = b2 + b3
Hence, we obtain
x1 = −α− 3b2 − b3; x2 = α + b2 + b3
X =
−α− 3b2 − b3α + b2 + b3
α
= α
−111
+
−3b2 − b3b2 + b3
0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Thus, we have two equations and one unknown,so one of thevariables let’s say x3, is equal to a parameter α, obtaining thesystem
x1 + α = −3b2 − b3x2 − α = b2 + b3
Hence, we obtain
x1 = −α− 3b2 − b3; x2 = α + b2 + b3
X =
−α− 3b2 − b3α + b2 + b3
α
= α
−111
+
−3b2 − b3b2 + b3
0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Thus, we have two equations and one unknown,so one of thevariables let’s say x3, is equal to a parameter α, obtaining thesystem
x1 + α = −3b2 − b3x2 − α = b2 + b3
Hence, we obtain
x1 = −α− 3b2 − b3; x2 = α + b2 + b3
X =
−α− 3b2 − b3α + b2 + b3
α
= α
−111
+
−3b2 − b3b2 + b3
0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Thus, we have two equations and one unknown,so one of thevariables let’s say x3, is equal to a parameter α, obtaining thesystem
x1 + α = −3b2 − b3x2 − α = b2 + b3
Hence, we obtain
x1 = −α− 3b2 − b3; x2 = α + b2 + b3
X =
−α− 3b2 − b3α + b2 + b3
α
=
α
−111
+
−3b2 − b3b2 + b3
0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Thus, we have two equations and one unknown,so one of thevariables let’s say x3, is equal to a parameter α, obtaining thesystem
x1 + α = −3b2 − b3x2 − α = b2 + b3
Hence, we obtain
x1 = −α− 3b2 − b3; x2 = α + b2 + b3
X =
−α− 3b2 − b3α + b2 + b3
α
= α
−111
+
−3b2 − b3b2 + b3
0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Linear Dependence and Independence .
A set of k vectors x(1), ..., x(k) is said to be linearly dependent ifthere exists a set of real or complex numbers c1, ...., ck , at leastone of which is nonzero, such that
c1x(1) + ...+ ckx(k) = 0
On the other hand, if the only set c1, ..., ck for which the aboveequation is satisfied is c1 = c2 = · · · = ck = 0,then the set ofvectors x(1), ..., x(k) is called linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Linear Dependence and Independence .
A set of k vectors x(1), ..., x(k) is said to be linearly dependent ifthere exists a set of real or complex numbers c1, ...., ck , at leastone of which is nonzero, such that
c1x(1) + ...+ ckx(k) = 0
On the other hand, if the only set c1, ..., ck for which the aboveequation is satisfied is c1 = c2 = · · · = ck = 0,then the set ofvectors x(1), ..., x(k) is called linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Linear Dependence and Independence .
A set of k vectors x(1), ..., x(k)
is said to be linearly dependent ifthere exists a set of real or complex numbers c1, ...., ck , at leastone of which is nonzero, such that
c1x(1) + ...+ ckx(k) = 0
On the other hand, if the only set c1, ..., ck for which the aboveequation is satisfied is c1 = c2 = · · · = ck = 0,then the set ofvectors x(1), ..., x(k) is called linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Linear Dependence and Independence .
A set of k vectors x(1), ..., x(k) is said to be linearly dependent
ifthere exists a set of real or complex numbers c1, ...., ck , at leastone of which is nonzero, such that
c1x(1) + ...+ ckx(k) = 0
On the other hand, if the only set c1, ..., ck for which the aboveequation is satisfied is c1 = c2 = · · · = ck = 0,then the set ofvectors x(1), ..., x(k) is called linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Linear Dependence and Independence .
A set of k vectors x(1), ..., x(k) is said to be linearly dependent ifthere exists a set of real or complex numbers c1, ...., ck ,
at leastone of which is nonzero, such that
c1x(1) + ...+ ckx(k) = 0
On the other hand, if the only set c1, ..., ck for which the aboveequation is satisfied is c1 = c2 = · · · = ck = 0,then the set ofvectors x(1), ..., x(k) is called linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Linear Dependence and Independence .
A set of k vectors x(1), ..., x(k) is said to be linearly dependent ifthere exists a set of real or complex numbers c1, ...., ck , at leastone of which is nonzero,
such that
c1x(1) + ...+ ckx(k) = 0
On the other hand, if the only set c1, ..., ck for which the aboveequation is satisfied is c1 = c2 = · · · = ck = 0,then the set ofvectors x(1), ..., x(k) is called linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Linear Dependence and Independence .
A set of k vectors x(1), ..., x(k) is said to be linearly dependent ifthere exists a set of real or complex numbers c1, ...., ck , at leastone of which is nonzero, such that
c1x(1) + ...+ ckx(k) = 0
On the other hand, if the only set c1, ..., ck for which the aboveequation is satisfied is c1 = c2 = · · · = ck = 0,then the set ofvectors x(1), ..., x(k) is called linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Linear Dependence and Independence .
A set of k vectors x(1), ..., x(k) is said to be linearly dependent ifthere exists a set of real or complex numbers c1, ...., ck , at leastone of which is nonzero, such that
c1x(1) + ...+ ckx(k) = 0
On the other hand, if the only set c1, ..., ck for which the aboveequation is satisfied is c1 = c2 = · · · = ck = 0,then the set ofvectors x(1), ..., x(k) is called linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Linear Dependence and Independence .
A set of k vectors x(1), ..., x(k) is said to be linearly dependent ifthere exists a set of real or complex numbers c1, ...., ck , at leastone of which is nonzero, such that
c1x(1) + ...+ ckx(k) = 0
On the other hand,
if the only set c1, ..., ck for which the aboveequation is satisfied is c1 = c2 = · · · = ck = 0,then the set ofvectors x(1), ..., x(k) is called linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Linear Dependence and Independence .
A set of k vectors x(1), ..., x(k) is said to be linearly dependent ifthere exists a set of real or complex numbers c1, ...., ck , at leastone of which is nonzero, such that
c1x(1) + ...+ ckx(k) = 0
On the other hand, if the only set c1, ..., ck
for which the aboveequation is satisfied is c1 = c2 = · · · = ck = 0,then the set ofvectors x(1), ..., x(k) is called linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Linear Dependence and Independence .
A set of k vectors x(1), ..., x(k) is said to be linearly dependent ifthere exists a set of real or complex numbers c1, ...., ck , at leastone of which is nonzero, such that
c1x(1) + ...+ ckx(k) = 0
On the other hand, if the only set c1, ..., ck for which the aboveequation is satisfied is
c1 = c2 = · · · = ck = 0,then the set ofvectors x(1), ..., x(k) is called linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Linear Dependence and Independence .
A set of k vectors x(1), ..., x(k) is said to be linearly dependent ifthere exists a set of real or complex numbers c1, ...., ck , at leastone of which is nonzero, such that
c1x(1) + ...+ ckx(k) = 0
On the other hand, if the only set c1, ..., ck for which the aboveequation is satisfied is c1 = c2 = · · · = ck = 0,then
the set ofvectors x(1), ..., x(k) is called linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Linear Dependence and Independence .
A set of k vectors x(1), ..., x(k) is said to be linearly dependent ifthere exists a set of real or complex numbers c1, ...., ck , at leastone of which is nonzero, such that
c1x(1) + ...+ ckx(k) = 0
On the other hand, if the only set c1, ..., ck for which the aboveequation is satisfied is c1 = c2 = · · · = ck = 0,then the set ofvectors x(1), ..., x(k)
is called linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Linear Dependence and Independence .
A set of k vectors x(1), ..., x(k) is said to be linearly dependent ifthere exists a set of real or complex numbers c1, ...., ck , at leastone of which is nonzero, such that
c1x(1) + ...+ ckx(k) = 0
On the other hand, if the only set c1, ..., ck for which the aboveequation is satisfied is c1 = c2 = · · · = ck = 0,then the set ofvectors x(1), ..., x(k) is called linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Consider now a set of n vectors, each of which has n components ,
x(1) =
x11x21
...xn1
; x(2) =
x12x22
...xn2
; · · · x(n) =
x1nx2n
...xnn
the above equation can be written as .
x11c1 + x12c2 + . . .+ x1ncnx21c1 + x22c2 + . . .+ x2ncn
......
xn1c1 + xn2c2 + . . .+ xnncn
= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Consider now a set of n vectors,
each of which has n components ,
x(1) =
x11x21
...xn1
; x(2) =
x12x22
...xn2
; · · · x(n) =
x1nx2n
...xnn
the above equation can be written as .
x11c1 + x12c2 + . . .+ x1ncnx21c1 + x22c2 + . . .+ x2ncn
......
xn1c1 + xn2c2 + . . .+ xnncn
= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Consider now a set of n vectors, each of which has n components ,
x(1) =
x11x21
...xn1
; x(2) =
x12x22
...xn2
; · · · x(n) =
x1nx2n
...xnn
the above equation can be written as .
x11c1 + x12c2 + . . .+ x1ncnx21c1 + x22c2 + . . .+ x2ncn
......
xn1c1 + xn2c2 + . . .+ xnncn
= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Consider now a set of n vectors, each of which has n components ,
x(1) =
x11x21
...xn1
;
x(2) =
x12x22
...xn2
; · · · x(n) =
x1nx2n
...xnn
the above equation can be written as .
x11c1 + x12c2 + . . .+ x1ncnx21c1 + x22c2 + . . .+ x2ncn
......
xn1c1 + xn2c2 + . . .+ xnncn
= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Consider now a set of n vectors, each of which has n components ,
x(1) =
x11x21
...xn1
; x(2) =
x12x22
...xn2
;
· · · x(n) =
x1nx2n
...xnn
the above equation can be written as .
x11c1 + x12c2 + . . .+ x1ncnx21c1 + x22c2 + . . .+ x2ncn
......
xn1c1 + xn2c2 + . . .+ xnncn
= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Consider now a set of n vectors, each of which has n components ,
x(1) =
x11x21
...xn1
; x(2) =
x12x22
...xn2
; · · ·
x(n) =
x1nx2n
...xnn
the above equation can be written as .
x11c1 + x12c2 + . . .+ x1ncnx21c1 + x22c2 + . . .+ x2ncn
......
xn1c1 + xn2c2 + . . .+ xnncn
= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Consider now a set of n vectors, each of which has n components ,
x(1) =
x11x21
...xn1
; x(2) =
x12x22
...xn2
; · · · x(n) =
x1nx2n
...xnn
the above equation can be written as .x11c1 + x12c2 + . . .+ x1ncnx21c1 + x22c2 + . . .+ x2ncn
......
xn1c1 + xn2c2 + . . .+ xnncn
= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Consider now a set of n vectors, each of which has n components ,
x(1) =
x11x21
...xn1
; x(2) =
x12x22
...xn2
; · · · x(n) =
x1nx2n
...xnn
the above equation can be written as .
x11c1 + x12c2 + . . .+ x1ncnx21c1 + x22c2 + . . .+ x2ncn
......
xn1c1 + xn2c2 + . . .+ xnncn
= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Consider now a set of n vectors, each of which has n components ,
x(1) =
x11x21
...xn1
; x(2) =
x12x22
...xn2
; · · · x(n) =
x1nx2n
...xnn
the above equation can be written as .
x11c1 + x12c2 + . . .+ x1ncnx21c1 + x22c2 + . . .+ x2ncn
......
xn1c1 + xn2c2 + . . .+ xnncn
= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
or equivalently
Xc = 0
If X is nonsingular (X−1 exists), then the only solution of is c = 0,but if X is singular (X−1 does not exist) there are nonzerosolutions.
Example 7.8
Determine wether the vectors are linearly indepent or not
x(1) =
12
− 1
; x(2) =
213
; x(3) =
− 41
− 11
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
or equivalently
Xc = 0
If X is nonsingular (X−1 exists), then the only solution of is c = 0,but if X is singular (X−1 does not exist) there are nonzerosolutions.
Example 7.8
Determine wether the vectors are linearly indepent or not
x(1) =
12
− 1
; x(2) =
213
; x(3) =
− 41
− 11
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
or equivalently
Xc = 0
If X is nonsingular (X−1 exists), then the only solution of is c = 0,but if X is singular (X−1 does not exist) there are nonzerosolutions.
Example 7.8
Determine wether the vectors are linearly indepent or not
x(1) =
12
− 1
; x(2) =
213
; x(3) =
− 41
− 11
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
or equivalently
Xc = 0
If X is nonsingular
(X−1 exists), then the only solution of is c = 0,but if X is singular (X−1 does not exist) there are nonzerosolutions.
Example 7.8
Determine wether the vectors are linearly indepent or not
x(1) =
12
− 1
; x(2) =
213
; x(3) =
− 41
− 11
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
or equivalently
Xc = 0
If X is nonsingular (X−1 exists), then
the only solution of is c = 0,but if X is singular (X−1 does not exist) there are nonzerosolutions.
Example 7.8
Determine wether the vectors are linearly indepent or not
x(1) =
12
− 1
; x(2) =
213
; x(3) =
− 41
− 11
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
or equivalently
Xc = 0
If X is nonsingular (X−1 exists), then the only solution of is c = 0,
but if X is singular (X−1 does not exist) there are nonzerosolutions.
Example 7.8
Determine wether the vectors are linearly indepent or not
x(1) =
12
− 1
; x(2) =
213
; x(3) =
− 41
− 11
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
or equivalently
Xc = 0
If X is nonsingular (X−1 exists), then the only solution of is c = 0,but if X is singular
(X−1 does not exist) there are nonzerosolutions.
Example 7.8
Determine wether the vectors are linearly indepent or not
x(1) =
12
− 1
; x(2) =
213
; x(3) =
− 41
− 11
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
or equivalently
Xc = 0
If X is nonsingular (X−1 exists), then the only solution of is c = 0,but if X is singular (X−1 does not exist)
there are nonzerosolutions.
Example 7.8
Determine wether the vectors are linearly indepent or not
x(1) =
12
− 1
; x(2) =
213
; x(3) =
− 41
− 11
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
or equivalently
Xc = 0
If X is nonsingular (X−1 exists), then the only solution of is c = 0,but if X is singular (X−1 does not exist) there are nonzerosolutions.
Example 7.8
Determine wether the vectors are linearly indepent or not
x(1) =
12
− 1
; x(2) =
213
; x(3) =
− 41
− 11
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
or equivalently
Xc = 0
If X is nonsingular (X−1 exists), then the only solution of is c = 0,but if X is singular (X−1 does not exist) there are nonzerosolutions.
Example 7.8
Determine wether the vectors are linearly indepent or not
x(1) =
12
− 1
; x(2) =
213
; x(3) =
− 41
− 11
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
or equivalently
Xc = 0
If X is nonsingular (X−1 exists), then the only solution of is c = 0,but if X is singular (X−1 does not exist) there are nonzerosolutions.
Example 7.8
Determine wether the vectors are linearly indepent or not
x(1) =
12
− 1
; x(2) =
213
; x(3) =
− 41
− 11
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
or equivalently
Xc = 0
If X is nonsingular (X−1 exists), then the only solution of is c = 0,but if X is singular (X−1 does not exist) there are nonzerosolutions.
Example 7.8
Determine wether the vectors are linearly indepent or not
x(1) =
12
− 1
;
x(2) =
213
; x(3) =
− 41
− 11
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
or equivalently
Xc = 0
If X is nonsingular (X−1 exists), then the only solution of is c = 0,but if X is singular (X−1 does not exist) there are nonzerosolutions.
Example 7.8
Determine wether the vectors are linearly indepent or not
x(1) =
12
− 1
; x(2) =
213
;
x(3) =
− 41
− 11
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
or equivalently
Xc = 0
If X is nonsingular (X−1 exists), then the only solution of is c = 0,but if X is singular (X−1 does not exist) there are nonzerosolutions.
Example 7.8
Determine wether the vectors are linearly indepent or not
x(1) =
12
− 1
; x(2) =
213
; x(3) =
− 41
− 11
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solution
To determine whether x (1), x (2), and x (3) are linearly dependent,we seek constants c1, c2, and c3 such that
c1x(1) + c2x(2) + c3x(3) = 0
written in the matrix form 1 2 42 1 1−1 3 −11
c1c2c3
= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solution
To determine whether x (1), x (2), and x (3) are linearly dependent,we seek constants c1, c2, and c3 such that
c1x(1) + c2x(2) + c3x(3) = 0
written in the matrix form 1 2 42 1 1−1 3 −11
c1c2c3
= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solution
To determine whether x (1), x (2), and x (3) are linearly dependent,
we seek constants c1, c2, and c3 such that
c1x(1) + c2x(2) + c3x(3) = 0
written in the matrix form 1 2 42 1 1−1 3 −11
c1c2c3
= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solution
To determine whether x (1), x (2), and x (3) are linearly dependent,we seek constants c1, c2, and c3 such that
c1x(1) + c2x(2) + c3x(3) = 0
written in the matrix form 1 2 42 1 1−1 3 −11
c1c2c3
= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solution
To determine whether x (1), x (2), and x (3) are linearly dependent,we seek constants c1, c2, and c3 such that
c1x(1) + c2x(2) + c3x(3) = 0
written in the matrix form 1 2 42 1 1−1 3 −11
c1c2c3
= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solution
To determine whether x (1), x (2), and x (3) are linearly dependent,we seek constants c1, c2, and c3 such that
c1x(1) + c2x(2) + c3x(3) = 0
written in the matrix form
1 2 42 1 1−1 3 −11
c1c2c3
= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solution
To determine whether x (1), x (2), and x (3) are linearly dependent,we seek constants c1, c2, and c3 such that
c1x(1) + c2x(2) + c3x(3) = 0
written in the matrix form 1 2 42 1 1−1 3 −11
c1c2c3
= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Using elementary row operations on the augmented matrix1 2 −4
∣∣∣ 0
2 1 1∣∣∣ 0
−1 3 −11∣∣∣ 0
(a) Add (−2) times the first row to the second row, and add thefirst row to the third row.
1 2 −4∣∣∣ 0
0 −3 9∣∣∣ 0
0 5 −15∣∣∣ 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Using elementary row operations on the augmented matrix
1 2 −4
∣∣∣ 0
2 1 1∣∣∣ 0
−1 3 −11∣∣∣ 0
(a) Add (−2) times the first row to the second row, and add thefirst row to the third row.
1 2 −4∣∣∣ 0
0 −3 9∣∣∣ 0
0 5 −15∣∣∣ 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Using elementary row operations on the augmented matrix1 2 −4
∣∣∣ 0
2 1 1∣∣∣ 0
−1 3 −11∣∣∣ 0
(a) Add (−2) times the first row to the second row, and add thefirst row to the third row.
1 2 −4∣∣∣ 0
0 −3 9∣∣∣ 0
0 5 −15∣∣∣ 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Using elementary row operations on the augmented matrix1 2 −4
∣∣∣ 0
2 1 1∣∣∣ 0
−1 3 −11∣∣∣ 0
(a) Add (−2) times the first row
to the second row, and add thefirst row to the third row.
1 2 −4∣∣∣ 0
0 −3 9∣∣∣ 0
0 5 −15∣∣∣ 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Using elementary row operations on the augmented matrix1 2 −4
∣∣∣ 0
2 1 1∣∣∣ 0
−1 3 −11∣∣∣ 0
(a) Add (−2) times the first row to the second row, and
add thefirst row to the third row.
1 2 −4∣∣∣ 0
0 −3 9∣∣∣ 0
0 5 −15∣∣∣ 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Using elementary row operations on the augmented matrix1 2 −4
∣∣∣ 0
2 1 1∣∣∣ 0
−1 3 −11∣∣∣ 0
(a) Add (−2) times the first row to the second row, and add thefirst row
to the third row.1 2 −4
∣∣∣ 0
0 −3 9∣∣∣ 0
0 5 −15∣∣∣ 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Using elementary row operations on the augmented matrix1 2 −4
∣∣∣ 0
2 1 1∣∣∣ 0
−1 3 −11∣∣∣ 0
(a) Add (−2) times the first row to the second row, and add thefirst row to the third row.
1 2 −4
∣∣∣ 0
0 −3 9∣∣∣ 0
0 5 −15∣∣∣ 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Using elementary row operations on the augmented matrix1 2 −4
∣∣∣ 0
2 1 1∣∣∣ 0
−1 3 −11∣∣∣ 0
(a) Add (−2) times the first row to the second row, and add thefirst row to the third row.
1 2 −4∣∣∣ 0
0 −3 9∣∣∣ 0
0 5 −15∣∣∣ 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(b) Divide the second row by (−3), then add (−5) times thesecond row to the third row.
1 2 −4∣∣∣ 0
0 1 −3∣∣∣ 0
0 0 0∣∣∣ 0
Thus we obtain the equivalent system(
c1 + 2c2 − 4c3 = 0c2 − 3c3 = 0
)= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(b) Divide the second row by (−3),
then add (−5) times thesecond row to the third row.
1 2 −4∣∣∣ 0
0 1 −3∣∣∣ 0
0 0 0∣∣∣ 0
Thus we obtain the equivalent system(
c1 + 2c2 − 4c3 = 0c2 − 3c3 = 0
)= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(b) Divide the second row by (−3), then add (−5) times thesecond row
to the third row.1 2 −4
∣∣∣ 0
0 1 −3∣∣∣ 0
0 0 0∣∣∣ 0
Thus we obtain the equivalent system(
c1 + 2c2 − 4c3 = 0c2 − 3c3 = 0
)= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(b) Divide the second row by (−3), then add (−5) times thesecond row to the third row.
1 2 −4
∣∣∣ 0
0 1 −3∣∣∣ 0
0 0 0∣∣∣ 0
Thus we obtain the equivalent system(
c1 + 2c2 − 4c3 = 0c2 − 3c3 = 0
)= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(b) Divide the second row by (−3), then add (−5) times thesecond row to the third row.
1 2 −4∣∣∣ 0
0 1 −3∣∣∣ 0
0 0 0∣∣∣ 0
Thus we obtain the equivalent system(c1 + 2c2 − 4c3 = 0
c2 − 3c3 = 0
)= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(b) Divide the second row by (−3), then add (−5) times thesecond row to the third row.
1 2 −4∣∣∣ 0
0 1 −3∣∣∣ 0
0 0 0∣∣∣ 0
Thus we obtain the equivalent system
(c1 + 2c2 − 4c3 = 0
c2 − 3c3 = 0
)= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(b) Divide the second row by (−3), then add (−5) times thesecond row to the third row.
1 2 −4∣∣∣ 0
0 1 −3∣∣∣ 0
0 0 0∣∣∣ 0
Thus we obtain the equivalent system(
c1 + 2c2 − 4c3 = 0c2 − 3c3 = 0
)= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Hence, we have 2 equations in 3 unknowns, so one of them, let’ssay c3 will be a free parameter (real number) α and the solution ofthe system is
c3 = α; c2 = 3c3 = 3α; c1 = −2c2 + 4c3 = −2c3 = −2α
c1c2c3
=
− 2α3αα
= α
− 231
Hence, there are infinitely solutions and the set of vectors islinearly dependent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Hence, we have 2 equations in 3 unknowns, so
one of them, let’ssay c3 will be a free parameter (real number) α and the solution ofthe system is
c3 = α; c2 = 3c3 = 3α; c1 = −2c2 + 4c3 = −2c3 = −2α
c1c2c3
=
− 2α3αα
= α
− 231
Hence, there are infinitely solutions and the set of vectors islinearly dependent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Hence, we have 2 equations in 3 unknowns, so one of them, let’ssay c3 will be a free parameter (real number) α and
the solution ofthe system is
c3 = α; c2 = 3c3 = 3α; c1 = −2c2 + 4c3 = −2c3 = −2α
c1c2c3
=
− 2α3αα
= α
− 231
Hence, there are infinitely solutions and the set of vectors islinearly dependent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Hence, we have 2 equations in 3 unknowns, so one of them, let’ssay c3 will be a free parameter (real number) α and the solution ofthe system is
c3 = α; c2 = 3c3 = 3α; c1 = −2c2 + 4c3 = −2c3 = −2α
c1c2c3
=
− 2α3αα
= α
− 231
Hence, there are infinitely solutions and the set of vectors islinearly dependent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Hence, we have 2 equations in 3 unknowns, so one of them, let’ssay c3 will be a free parameter (real number) α and the solution ofthe system is
c3 = α;
c2 = 3c3 = 3α; c1 = −2c2 + 4c3 = −2c3 = −2α
c1c2c3
=
− 2α3αα
= α
− 231
Hence, there are infinitely solutions and the set of vectors islinearly dependent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Hence, we have 2 equations in 3 unknowns, so one of them, let’ssay c3 will be a free parameter (real number) α and the solution ofthe system is
c3 = α; c2 = 3c3 = 3α;
c1 = −2c2 + 4c3 = −2c3 = −2α
c1c2c3
=
− 2α3αα
= α
− 231
Hence, there are infinitely solutions and the set of vectors islinearly dependent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Hence, we have 2 equations in 3 unknowns, so one of them, let’ssay c3 will be a free parameter (real number) α and the solution ofthe system is
c3 = α; c2 = 3c3 = 3α; c1 = −2c2 + 4c3 = −2c3 = −2α
c1c2c3
=
− 2α3αα
= α
− 231
Hence, there are infinitely solutions and the set of vectors islinearly dependent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Hence, we have 2 equations in 3 unknowns, so one of them, let’ssay c3 will be a free parameter (real number) α and the solution ofthe system is
c3 = α; c2 = 3c3 = 3α; c1 = −2c2 + 4c3 = −2c3 = −2α
c1c2c3
=
− 2α3αα
=
α
− 231
Hence, there are infinitely solutions and the set of vectors islinearly dependent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Hence, we have 2 equations in 3 unknowns, so one of them, let’ssay c3 will be a free parameter (real number) α and the solution ofthe system is
c3 = α; c2 = 3c3 = 3α; c1 = −2c2 + 4c3 = −2c3 = −2α
c1c2c3
=
− 2α3αα
= α
− 231
Hence, there are infinitely solutions and the set of vectors islinearly dependent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Hence, we have 2 equations in 3 unknowns, so one of them, let’ssay c3 will be a free parameter (real number) α and the solution ofthe system is
c3 = α; c2 = 3c3 = 3α; c1 = −2c2 + 4c3 = −2c3 = −2α
c1c2c3
=
− 2α3αα
= α
− 231
Hence, there are infinitely solutions and
the set of vectors islinearly dependent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Hence, we have 2 equations in 3 unknowns, so one of them, let’ssay c3 will be a free parameter (real number) α and the solution ofthe system is
c3 = α; c2 = 3c3 = 3α; c1 = −2c2 + 4c3 = −2c3 = −2α
c1c2c3
=
− 2α3αα
= α
− 231
Hence, there are infinitely solutions and the set of vectors islinearly dependent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Determinants
Associated to every n × n matrix A there is a real number calledthe determinant of A denoted by |A| or det(A) and definedinductivly as follows
n = 1 A = a11 |A| = a11
n = 2 A =
(a11 a12a21 a22
)|A| =
∣∣∣∣a11 a12a21 a22
∣∣∣∣ = a11a22 − a12a21
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Determinants
Associated to every n × n matrix A there is a real number calledthe determinant of A denoted by |A| or det(A) and definedinductivly as follows
n = 1 A = a11 |A| = a11
n = 2 A =
(a11 a12a21 a22
)|A| =
∣∣∣∣a11 a12a21 a22
∣∣∣∣ = a11a22 − a12a21
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Determinants
Associated to every n × n matrix A
there is a real number calledthe determinant of A denoted by |A| or det(A) and definedinductivly as follows
n = 1 A = a11 |A| = a11
n = 2 A =
(a11 a12a21 a22
)|A| =
∣∣∣∣a11 a12a21 a22
∣∣∣∣ = a11a22 − a12a21
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Determinants
Associated to every n × n matrix A there is a real number calledthe determinant of A
denoted by |A| or det(A) and definedinductivly as follows
n = 1 A = a11 |A| = a11
n = 2 A =
(a11 a12a21 a22
)|A| =
∣∣∣∣a11 a12a21 a22
∣∣∣∣ = a11a22 − a12a21
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Determinants
Associated to every n × n matrix A there is a real number calledthe determinant of A denoted by |A| or det(A) and
definedinductivly as follows
n = 1 A = a11 |A| = a11
n = 2 A =
(a11 a12a21 a22
)|A| =
∣∣∣∣a11 a12a21 a22
∣∣∣∣ = a11a22 − a12a21
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Determinants
Associated to every n × n matrix A there is a real number calledthe determinant of A denoted by |A| or det(A) and definedinductivly as follows
n = 1 A = a11 |A| = a11
n = 2 A =
(a11 a12a21 a22
)|A| =
∣∣∣∣a11 a12a21 a22
∣∣∣∣ = a11a22 − a12a21
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Determinants
Associated to every n × n matrix A there is a real number calledthe determinant of A denoted by |A| or det(A) and definedinductivly as follows
n = 1 A = a11 |A| = a11
n = 2 A =
(a11 a12a21 a22
)|A| =
∣∣∣∣a11 a12a21 a22
∣∣∣∣ = a11a22 − a12a21
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Determinants
Associated to every n × n matrix A there is a real number calledthe determinant of A denoted by |A| or det(A) and definedinductivly as follows
n = 1
A = a11 |A| = a11
n = 2 A =
(a11 a12a21 a22
)|A| =
∣∣∣∣a11 a12a21 a22
∣∣∣∣ = a11a22 − a12a21
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Determinants
Associated to every n × n matrix A there is a real number calledthe determinant of A denoted by |A| or det(A) and definedinductivly as follows
n = 1 A = a11
|A| = a11
n = 2 A =
(a11 a12a21 a22
)|A| =
∣∣∣∣a11 a12a21 a22
∣∣∣∣ = a11a22 − a12a21
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Determinants
Associated to every n × n matrix A there is a real number calledthe determinant of A denoted by |A| or det(A) and definedinductivly as follows
n = 1 A = a11 |A| = a11
n = 2 A =
(a11 a12a21 a22
)|A| =
∣∣∣∣a11 a12a21 a22
∣∣∣∣ = a11a22 − a12a21
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Determinants
Associated to every n × n matrix A there is a real number calledthe determinant of A denoted by |A| or det(A) and definedinductivly as follows
n = 1 A = a11 |A| = a11
n = 2
A =
(a11 a12a21 a22
)|A| =
∣∣∣∣a11 a12a21 a22
∣∣∣∣ = a11a22 − a12a21
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Determinants
Associated to every n × n matrix A there is a real number calledthe determinant of A denoted by |A| or det(A) and definedinductivly as follows
n = 1 A = a11 |A| = a11
n = 2 A =
(a11 a12a21 a22
)
|A| =
∣∣∣∣a11 a12a21 a22
∣∣∣∣ = a11a22 − a12a21
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Determinants
Associated to every n × n matrix A there is a real number calledthe determinant of A denoted by |A| or det(A) and definedinductivly as follows
n = 1 A = a11 |A| = a11
n = 2 A =
(a11 a12a21 a22
)|A| =
∣∣∣∣a11 a12a21 a22
∣∣∣∣ =
a11a22 − a12a21
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Determinants
Associated to every n × n matrix A there is a real number calledthe determinant of A denoted by |A| or det(A) and definedinductivly as follows
n = 1 A = a11 |A| = a11
n = 2 A =
(a11 a12a21 a22
)|A| =
∣∣∣∣a11 a12a21 a22
∣∣∣∣ = a11a22 − a12a21
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
n = 3 A =
a11 a12 a13a21 a22 a23a31 a32 a33
|A| =
∣∣∣∣∣∣a11 a12 a13a21 a22 a23a31 a32 a33
∣∣∣∣∣∣ =
a11
∣∣∣∣a22 a23a32 a33
∣∣∣∣− a12
∣∣∣∣a21 a23a31 a33
∣∣∣∣+ a13
∣∣∣∣a11 a12a31 a32
∣∣∣∣ = a11a22a33+
a12a23a31 + a13a21a32 − a31a22a13 − a32a23a11 − a33a21a12 =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
n = 3
A =
a11 a12 a13a21 a22 a23a31 a32 a33
|A| =
∣∣∣∣∣∣a11 a12 a13a21 a22 a23a31 a32 a33
∣∣∣∣∣∣ =
a11
∣∣∣∣a22 a23a32 a33
∣∣∣∣− a12
∣∣∣∣a21 a23a31 a33
∣∣∣∣+ a13
∣∣∣∣a11 a12a31 a32
∣∣∣∣ = a11a22a33+
a12a23a31 + a13a21a32 − a31a22a13 − a32a23a11 − a33a21a12 =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
n = 3 A =
a11 a12 a13a21 a22 a23a31 a32 a33
|A| =
∣∣∣∣∣∣a11 a12 a13a21 a22 a23a31 a32 a33
∣∣∣∣∣∣ =
a11
∣∣∣∣a22 a23a32 a33
∣∣∣∣− a12
∣∣∣∣a21 a23a31 a33
∣∣∣∣+ a13
∣∣∣∣a11 a12a31 a32
∣∣∣∣ = a11a22a33+
a12a23a31 + a13a21a32 − a31a22a13 − a32a23a11 − a33a21a12 =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
n = 3 A =
a11 a12 a13a21 a22 a23a31 a32 a33
|A| =
∣∣∣∣∣∣a11 a12 a13a21 a22 a23a31 a32 a33
∣∣∣∣∣∣ =
a11
∣∣∣∣a22 a23a32 a33
∣∣∣∣− a12
∣∣∣∣a21 a23a31 a33
∣∣∣∣+ a13
∣∣∣∣a11 a12a31 a32
∣∣∣∣ = a11a22a33+
a12a23a31 + a13a21a32 − a31a22a13 − a32a23a11 − a33a21a12 =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
n = 3 A =
a11 a12 a13a21 a22 a23a31 a32 a33
|A| =
∣∣∣∣∣∣a11 a12 a13a21 a22 a23a31 a32 a33
∣∣∣∣∣∣ =
a11
∣∣∣∣a22 a23a32 a33
∣∣∣∣− a12
∣∣∣∣a21 a23a31 a33
∣∣∣∣+ a13
∣∣∣∣a11 a12a31 a32
∣∣∣∣ = a11a22a33+
a12a23a31 + a13a21a32 − a31a22a13 − a32a23a11 − a33a21a12 =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
n = 3 A =
a11 a12 a13a21 a22 a23a31 a32 a33
|A| =
∣∣∣∣∣∣a11 a12 a13a21 a22 a23a31 a32 a33
∣∣∣∣∣∣ =
a11
∣∣∣∣a22 a23a32 a33
∣∣∣∣−
a12
∣∣∣∣a21 a23a31 a33
∣∣∣∣+ a13
∣∣∣∣a11 a12a31 a32
∣∣∣∣ = a11a22a33+
a12a23a31 + a13a21a32 − a31a22a13 − a32a23a11 − a33a21a12 =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
n = 3 A =
a11 a12 a13a21 a22 a23a31 a32 a33
|A| =
∣∣∣∣∣∣a11 a12 a13a21 a22 a23a31 a32 a33
∣∣∣∣∣∣ =
a11
∣∣∣∣a22 a23a32 a33
∣∣∣∣− a12
∣∣∣∣a21 a23a31 a33
∣∣∣∣+ a13
∣∣∣∣a11 a12a31 a32
∣∣∣∣ = a11a22a33+
a12a23a31 + a13a21a32 − a31a22a13 − a32a23a11 − a33a21a12 =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
n = 3 A =
a11 a12 a13a21 a22 a23a31 a32 a33
|A| =
∣∣∣∣∣∣a11 a12 a13a21 a22 a23a31 a32 a33
∣∣∣∣∣∣ =
a11
∣∣∣∣a22 a23a32 a33
∣∣∣∣− a12
∣∣∣∣a21 a23a31 a33
∣∣∣∣+
a13
∣∣∣∣a11 a12a31 a32
∣∣∣∣ = a11a22a33+
a12a23a31 + a13a21a32 − a31a22a13 − a32a23a11 − a33a21a12 =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
n = 3 A =
a11 a12 a13a21 a22 a23a31 a32 a33
|A| =
∣∣∣∣∣∣a11 a12 a13a21 a22 a23a31 a32 a33
∣∣∣∣∣∣ =
a11
∣∣∣∣a22 a23a32 a33
∣∣∣∣− a12
∣∣∣∣a21 a23a31 a33
∣∣∣∣+ a13
∣∣∣∣a11 a12a31 a32
∣∣∣∣ = a11a22a33+
a12a23a31 + a13a21a32 − a31a22a13 − a32a23a11 − a33a21a12 =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
n = 3 A =
a11 a12 a13a21 a22 a23a31 a32 a33
|A| =
∣∣∣∣∣∣a11 a12 a13a21 a22 a23a31 a32 a33
∣∣∣∣∣∣ =
a11
∣∣∣∣a22 a23a32 a33
∣∣∣∣− a12
∣∣∣∣a21 a23a31 a33
∣∣∣∣+ a13
∣∣∣∣a11 a12a31 a32
∣∣∣∣ = a11a22a33+
a12a23a31 + a13a21a32 − a31a22a13 − a32a23a11 − a33a21a12 =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
n = 3 A =
a11 a12 a13a21 a22 a23a31 a32 a33
|A| =
∣∣∣∣∣∣a11 a12 a13a21 a22 a23a31 a32 a33
∣∣∣∣∣∣ =
a11
∣∣∣∣a22 a23a32 a33
∣∣∣∣− a12
∣∣∣∣a21 a23a31 a33
∣∣∣∣+ a13
∣∣∣∣a11 a12a31 a32
∣∣∣∣ = a11a22a33+
a12a23a31 +
a13a21a32 − a31a22a13 − a32a23a11 − a33a21a12 =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
n = 3 A =
a11 a12 a13a21 a22 a23a31 a32 a33
|A| =
∣∣∣∣∣∣a11 a12 a13a21 a22 a23a31 a32 a33
∣∣∣∣∣∣ =
a11
∣∣∣∣a22 a23a32 a33
∣∣∣∣− a12
∣∣∣∣a21 a23a31 a33
∣∣∣∣+ a13
∣∣∣∣a11 a12a31 a32
∣∣∣∣ = a11a22a33+
a12a23a31 + a13a21a32 −
a31a22a13 − a32a23a11 − a33a21a12 =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
n = 3 A =
a11 a12 a13a21 a22 a23a31 a32 a33
|A| =
∣∣∣∣∣∣a11 a12 a13a21 a22 a23a31 a32 a33
∣∣∣∣∣∣ =
a11
∣∣∣∣a22 a23a32 a33
∣∣∣∣− a12
∣∣∣∣a21 a23a31 a33
∣∣∣∣+ a13
∣∣∣∣a11 a12a31 a32
∣∣∣∣ = a11a22a33+
a12a23a31 + a13a21a32 − a31a22a13 −
a32a23a11 − a33a21a12 =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
n = 3 A =
a11 a12 a13a21 a22 a23a31 a32 a33
|A| =
∣∣∣∣∣∣a11 a12 a13a21 a22 a23a31 a32 a33
∣∣∣∣∣∣ =
a11
∣∣∣∣a22 a23a32 a33
∣∣∣∣− a12
∣∣∣∣a21 a23a31 a33
∣∣∣∣+ a13
∣∣∣∣a11 a12a31 a32
∣∣∣∣ = a11a22a33+
a12a23a31 + a13a21a32 − a31a22a13 − a32a23a11 −
a33a21a12 =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
n = 3 A =
a11 a12 a13a21 a22 a23a31 a32 a33
|A| =
∣∣∣∣∣∣a11 a12 a13a21 a22 a23a31 a32 a33
∣∣∣∣∣∣ =
a11
∣∣∣∣a22 a23a32 a33
∣∣∣∣− a12
∣∣∣∣a21 a23a31 a33
∣∣∣∣+ a13
∣∣∣∣a11 a12a31 a32
∣∣∣∣ = a11a22a33+
a12a23a31 + a13a21a32 − a31a22a13 − a32a23a11 − a33a21a12 =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
a11a22a33 + a12a23a31+a13a21a32 − a31a22a13−a32a23a11 − a33a21a12
=
∣∣∣∣∣∣∣∣a11 a12 a13 a11 a12a21 a22 a23 a21 a22a31 a32 a33 a31 a32
− − − + + +
∣∣∣∣∣∣∣∣Now, for n ≥ 4, if let M1j be the corresponding minors to the firstrow, then we have
|A| =n∑
j=1
(−1)1+ja1jM1j
or using any fix row i
|A| =n∑
j=1
(−1)i+jaijMij
or using any fix column j
|A| =n∑
i=1
(−1)i+jaijMij
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
a11a22a33 + a12a23a31+
a13a21a32 − a31a22a13−a32a23a11 − a33a21a12
=
∣∣∣∣∣∣∣∣a11 a12 a13 a11 a12a21 a22 a23 a21 a22a31 a32 a33 a31 a32
− − − + + +
∣∣∣∣∣∣∣∣Now, for n ≥ 4, if let M1j be the corresponding minors to the firstrow, then we have
|A| =n∑
j=1
(−1)1+ja1jM1j
or using any fix row i
|A| =n∑
j=1
(−1)i+jaijMij
or using any fix column j
|A| =n∑
i=1
(−1)i+jaijMij
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
a11a22a33 + a12a23a31+a13a21a32 − a31a22a13−
a32a23a11 − a33a21a12
=
∣∣∣∣∣∣∣∣a11 a12 a13 a11 a12a21 a22 a23 a21 a22a31 a32 a33 a31 a32
− − − + + +
∣∣∣∣∣∣∣∣Now, for n ≥ 4, if let M1j be the corresponding minors to the firstrow, then we have
|A| =n∑
j=1
(−1)1+ja1jM1j
or using any fix row i
|A| =n∑
j=1
(−1)i+jaijMij
or using any fix column j
|A| =n∑
i=1
(−1)i+jaijMij
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
a11a22a33 + a12a23a31+a13a21a32 − a31a22a13−a32a23a11 −
a33a21a12
=
∣∣∣∣∣∣∣∣a11 a12 a13 a11 a12a21 a22 a23 a21 a22a31 a32 a33 a31 a32
− − − + + +
∣∣∣∣∣∣∣∣Now, for n ≥ 4, if let M1j be the corresponding minors to the firstrow, then we have
|A| =n∑
j=1
(−1)1+ja1jM1j
or using any fix row i
|A| =n∑
j=1
(−1)i+jaijMij
or using any fix column j
|A| =n∑
i=1
(−1)i+jaijMij
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
a11a22a33 + a12a23a31+a13a21a32 − a31a22a13−a32a23a11 − a33a21a12
=
∣∣∣∣∣∣∣∣a11 a12 a13 a11 a12a21 a22 a23 a21 a22a31 a32 a33 a31 a32
− − − + + +
∣∣∣∣∣∣∣∣
Now, for n ≥ 4, if let M1j be the corresponding minors to the firstrow, then we have
|A| =n∑
j=1
(−1)1+ja1jM1j
or using any fix row i
|A| =n∑
j=1
(−1)i+jaijMij
or using any fix column j
|A| =n∑
i=1
(−1)i+jaijMij
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
a11a22a33 + a12a23a31+a13a21a32 − a31a22a13−a32a23a11 − a33a21a12
=
∣∣∣∣∣∣∣∣a11 a12 a13 a11 a12a21 a22 a23 a21 a22a31 a32 a33 a31 a32
− − − + + +
∣∣∣∣∣∣∣∣Now, for n ≥ 4, if let M1j be the corresponding minors to the firstrow, then we have
|A| =n∑
j=1
(−1)1+ja1jM1j
or using any fix row i
|A| =n∑
j=1
(−1)i+jaijMij
or using any fix column j
|A| =n∑
i=1
(−1)i+jaijMij
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
a11a22a33 + a12a23a31+a13a21a32 − a31a22a13−a32a23a11 − a33a21a12
=
∣∣∣∣∣∣∣∣a11 a12 a13 a11 a12a21 a22 a23 a21 a22a31 a32 a33 a31 a32
− − − + + +
∣∣∣∣∣∣∣∣Now, for n ≥ 4, if let M1j be the corresponding minors to the firstrow, then we have
|A| =n∑
j=1
(−1)1+ja1jM1j
or using any fix row i
|A| =n∑
j=1
(−1)i+jaijMij
or using any fix column j
|A| =n∑
i=1
(−1)i+jaijMij
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
a11a22a33 + a12a23a31+a13a21a32 − a31a22a13−a32a23a11 − a33a21a12
=
∣∣∣∣∣∣∣∣a11 a12 a13 a11 a12a21 a22 a23 a21 a22a31 a32 a33 a31 a32
− − − + + +
∣∣∣∣∣∣∣∣Now, for n ≥ 4, if let M1j be the corresponding minors to the firstrow, then we have
|A| =n∑
j=1
(−1)1+ja1jM1j
or using any fix row i
|A| =n∑
j=1
(−1)i+jaijMij
or using any fix column j
|A| =n∑
i=1
(−1)i+jaijMij
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
a11a22a33 + a12a23a31+a13a21a32 − a31a22a13−a32a23a11 − a33a21a12
=
∣∣∣∣∣∣∣∣a11 a12 a13 a11 a12a21 a22 a23 a21 a22a31 a32 a33 a31 a32
− − − + + +
∣∣∣∣∣∣∣∣Now, for n ≥ 4, if let M1j be the corresponding minors to the firstrow, then we have
|A| =n∑
j=1
(−1)1+ja1jM1j
or using any fix row i
|A| =n∑
j=1
(−1)i+jaijMij
or using any fix column j
|A| =n∑
i=1
(−1)i+jaijMij
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
a11a22a33 + a12a23a31+a13a21a32 − a31a22a13−a32a23a11 − a33a21a12
=
∣∣∣∣∣∣∣∣a11 a12 a13 a11 a12a21 a22 a23 a21 a22a31 a32 a33 a31 a32
− − − + + +
∣∣∣∣∣∣∣∣Now, for n ≥ 4, if let M1j be the corresponding minors to the firstrow, then we have
|A| =n∑
j=1
(−1)1+ja1jM1j
or using any fix row i
|A| =n∑
j=1
(−1)i+jaijMij
or using any fix column j
|A| =n∑
i=1
(−1)i+jaijMij
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
a11a22a33 + a12a23a31+a13a21a32 − a31a22a13−a32a23a11 − a33a21a12
=
∣∣∣∣∣∣∣∣a11 a12 a13 a11 a12a21 a22 a23 a21 a22a31 a32 a33 a31 a32
− − − + + +
∣∣∣∣∣∣∣∣Now, for n ≥ 4, if let M1j be the corresponding minors to the firstrow, then we have
|A| =n∑
j=1
(−1)1+ja1jM1j
or using any fix row i
|A| =n∑
j=1
(−1)i+jaijMij
or using any fix column j
|A| =n∑
i=1
(−1)i+jaijMij
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.1
Given an n × n matrix A, if B is an n × n matrix obtained from Aby
1) Adding a multiple of the ith row (column) to the jth row then|B| = |A|
2) Interchanging two consecutive rows (columns), then |B| = −|A|
3) Multiplying a row (column) by a nonzero scalar α then|B| = α|A|
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.1
Given an n × n matrix A, if B is an n × n matrix obtained from Aby
1) Adding a multiple of the ith row (column) to the jth row then|B| = |A|
2) Interchanging two consecutive rows (columns), then |B| = −|A|
3) Multiplying a row (column) by a nonzero scalar α then|B| = α|A|
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.1
Given an n × n matrix A,
if B is an n × n matrix obtained from Aby
1) Adding a multiple of the ith row (column) to the jth row then|B| = |A|
2) Interchanging two consecutive rows (columns), then |B| = −|A|
3) Multiplying a row (column) by a nonzero scalar α then|B| = α|A|
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.1
Given an n × n matrix A, if B is an n × n matrix obtained from Aby
1) Adding a multiple of the ith row (column) to the jth row then|B| = |A|
2) Interchanging two consecutive rows (columns), then |B| = −|A|
3) Multiplying a row (column) by a nonzero scalar α then|B| = α|A|
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.1
Given an n × n matrix A, if B is an n × n matrix obtained from Aby
1) Adding a multiple of the ith row (column) to the jth row then
|B| = |A|
2) Interchanging two consecutive rows (columns), then |B| = −|A|
3) Multiplying a row (column) by a nonzero scalar α then|B| = α|A|
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.1
Given an n × n matrix A, if B is an n × n matrix obtained from Aby
1) Adding a multiple of the ith row (column) to the jth row then|B| = |A|
2) Interchanging two consecutive rows (columns), then |B| = −|A|
3) Multiplying a row (column) by a nonzero scalar α then|B| = α|A|
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.1
Given an n × n matrix A, if B is an n × n matrix obtained from Aby
1) Adding a multiple of the ith row (column) to the jth row then|B| = |A|
2) Interchanging two consecutive rows (columns), then
|B| = −|A|
3) Multiplying a row (column) by a nonzero scalar α then|B| = α|A|
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.1
Given an n × n matrix A, if B is an n × n matrix obtained from Aby
1) Adding a multiple of the ith row (column) to the jth row then|B| = |A|
2) Interchanging two consecutive rows (columns), then |B| = −|A|
3) Multiplying a row (column) by a nonzero scalar α then|B| = α|A|
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.1
Given an n × n matrix A, if B is an n × n matrix obtained from Aby
1) Adding a multiple of the ith row (column) to the jth row then|B| = |A|
2) Interchanging two consecutive rows (columns), then |B| = −|A|
3) Multiplying a row (column) by a nonzero scalar α then
|B| = α|A|
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.1
Given an n × n matrix A, if B is an n × n matrix obtained from Aby
1) Adding a multiple of the ith row (column) to the jth row then|B| = |A|
2) Interchanging two consecutive rows (columns), then |B| = −|A|
3) Multiplying a row (column) by a nonzero scalar α then|B| = α|A|
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.2
1) |AT | = |A|
2) |AB| = |A||B|
3) If A has a row (column) of zeros, then |A| = 0
4) If A has a two identical rows (columns), then |A| = 0
5) If two rows (columns) of A are proportional, then |A| = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.2
1) |AT | = |A|
2) |AB| = |A||B|
3) If A has a row (column) of zeros, then |A| = 0
4) If A has a two identical rows (columns), then |A| = 0
5) If two rows (columns) of A are proportional, then |A| = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.2
1) |AT | = |A|
2) |AB| = |A||B|
3) If A has a row (column) of zeros, then |A| = 0
4) If A has a two identical rows (columns), then |A| = 0
5) If two rows (columns) of A are proportional, then |A| = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.2
1) |AT | = |A|
2) |AB| = |A||B|
3) If A has a row (column) of zeros, then |A| = 0
4) If A has a two identical rows (columns), then |A| = 0
5) If two rows (columns) of A are proportional, then |A| = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.2
1) |AT | = |A|
2) |AB| = |A||B|
3) If A has a row (column) of zeros, then
|A| = 0
4) If A has a two identical rows (columns), then |A| = 0
5) If two rows (columns) of A are proportional, then |A| = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.2
1) |AT | = |A|
2) |AB| = |A||B|
3) If A has a row (column) of zeros, then |A| = 0
4) If A has a two identical rows (columns), then |A| = 0
5) If two rows (columns) of A are proportional, then |A| = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.2
1) |AT | = |A|
2) |AB| = |A||B|
3) If A has a row (column) of zeros, then |A| = 0
4) If A has a two identical rows (columns), then
|A| = 0
5) If two rows (columns) of A are proportional, then |A| = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.2
1) |AT | = |A|
2) |AB| = |A||B|
3) If A has a row (column) of zeros, then |A| = 0
4) If A has a two identical rows (columns), then |A| = 0
5) If two rows (columns) of A are proportional, then |A| = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.2
1) |AT | = |A|
2) |AB| = |A||B|
3) If A has a row (column) of zeros, then |A| = 0
4) If A has a two identical rows (columns), then |A| = 0
5) If two rows (columns) of A are proportional, then
|A| = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.2
1) |AT | = |A|
2) |AB| = |A||B|
3) If A has a row (column) of zeros, then |A| = 0
4) If A has a two identical rows (columns), then |A| = 0
5) If two rows (columns) of A are proportional, then |A| = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
6) If A is an upper (lower) triangular matrix, then|A| = a11a22a33 · · · ann
Example 7.9
Find the following determinant of the matrix
A =
1 −1 2 4−1 3 −2 10 2 1 0−3 1 1 −1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
6) If A is an upper (lower) triangular matrix, then
|A| = a11a22a33 · · · ann
Example 7.9
Find the following determinant of the matrix
A =
1 −1 2 4−1 3 −2 10 2 1 0−3 1 1 −1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
6) If A is an upper (lower) triangular matrix, then|A| = a11a22a33 · · · ann
Example 7.9
Find the following determinant of the matrix
A =
1 −1 2 4−1 3 −2 10 2 1 0−3 1 1 −1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
6) If A is an upper (lower) triangular matrix, then|A| = a11a22a33 · · · ann
Example 7.9
Find the following determinant of the matrix
A =
1 −1 2 4−1 3 −2 10 2 1 0−3 1 1 −1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
6) If A is an upper (lower) triangular matrix, then|A| = a11a22a33 · · · ann
Example 7.9
Find the following determinant of the matrix
A =
1 −1 2 4−1 3 −2 10 2 1 0−3 1 1 −1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
6) If A is an upper (lower) triangular matrix, then|A| = a11a22a33 · · · ann
Example 7.9
Find the following determinant of the matrix
A =
1 −1 2 4−1 3 −2 10 2 1 0−3 1 1 −1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solution
|A| =
∣∣∣∣∣∣∣∣1 −1 2 4−1 3 −2 10 2 1 0−3 1 1 −1
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 2 1 00 −2 7 11
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 0 1 −50 0 7 16
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 0 1 −50 0 0 51
∣∣∣∣∣∣∣∣ = (1)(2)(1)(51) = 102
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solution
|A| =
∣∣∣∣∣∣∣∣1 −1 2 4−1 3 −2 10 2 1 0−3 1 1 −1
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 2 1 00 −2 7 11
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 0 1 −50 0 7 16
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 0 1 −50 0 0 51
∣∣∣∣∣∣∣∣ = (1)(2)(1)(51) = 102
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solution
|A| =
∣∣∣∣∣∣∣∣1 −1 2 4−1 3 −2 10 2 1 0−3 1 1 −1
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 2 1 00 −2 7 11
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 0 1 −50 0 7 16
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 0 1 −50 0 0 51
∣∣∣∣∣∣∣∣ = (1)(2)(1)(51) = 102
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solution
|A| =
∣∣∣∣∣∣∣∣1 −1 2 4−1 3 −2 10 2 1 0−3 1 1 −1
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 2 1 00 −2 7 11
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 0 1 −50 0 7 16
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 0 1 −50 0 0 51
∣∣∣∣∣∣∣∣ = (1)(2)(1)(51) = 102
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solution
|A| =
∣∣∣∣∣∣∣∣1 −1 2 4−1 3 −2 10 2 1 0−3 1 1 −1
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 2 1 00 −2 7 11
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 0 1 −50 0 7 16
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 0 1 −50 0 0 51
∣∣∣∣∣∣∣∣ = (1)(2)(1)(51) = 102
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solution
|A| =
∣∣∣∣∣∣∣∣1 −1 2 4−1 3 −2 10 2 1 0−3 1 1 −1
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 2 1 00 −2 7 11
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 0 1 −50 0 7 16
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 0 1 −50 0 0 51
∣∣∣∣∣∣∣∣ =
(1)(2)(1)(51) = 102
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solution
|A| =
∣∣∣∣∣∣∣∣1 −1 2 4−1 3 −2 10 2 1 0−3 1 1 −1
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 2 1 00 −2 7 11
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 0 1 −50 0 7 16
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 0 1 −50 0 0 51
∣∣∣∣∣∣∣∣ = (1)(2)(1)(51) =
102
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solution
|A| =
∣∣∣∣∣∣∣∣1 −1 2 4−1 3 −2 10 2 1 0−3 1 1 −1
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 2 1 00 −2 7 11
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 0 1 −50 0 7 16
∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣1 −1 2 40 2 0 50 0 1 −50 0 0 51
∣∣∣∣∣∣∣∣ = (1)(2)(1)(51) = 102
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.3
A matrix A is singular ⇐⇒ |A| = 0 ⇐⇒ Ax = 0 has anonzero solution ⇐⇒ Columns of A are linearly dependent.
Eigenvalues and Eigenvectors.
The equation
Ax = y
can be viewed as a linear transformation that maps (or transforms)a given vector x into a new vector x.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.3
A matrix A is singular ⇐⇒ |A| = 0 ⇐⇒ Ax = 0 has anonzero solution ⇐⇒ Columns of A are linearly dependent.
Eigenvalues and Eigenvectors.
The equation
Ax = y
can be viewed as a linear transformation that maps (or transforms)a given vector x into a new vector x.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.3
A matrix A is singular ⇐⇒
|A| = 0 ⇐⇒ Ax = 0 has anonzero solution ⇐⇒ Columns of A are linearly dependent.
Eigenvalues and Eigenvectors.
The equation
Ax = y
can be viewed as a linear transformation that maps (or transforms)a given vector x into a new vector x.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.3
A matrix A is singular ⇐⇒ |A| = 0 ⇐⇒
Ax = 0 has anonzero solution ⇐⇒ Columns of A are linearly dependent.
Eigenvalues and Eigenvectors.
The equation
Ax = y
can be viewed as a linear transformation that maps (or transforms)a given vector x into a new vector x.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.3
A matrix A is singular ⇐⇒ |A| = 0 ⇐⇒ Ax = 0 has anonzero solution ⇐⇒
Columns of A are linearly dependent.
Eigenvalues and Eigenvectors.
The equation
Ax = y
can be viewed as a linear transformation that maps (or transforms)a given vector x into a new vector x.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.3
A matrix A is singular ⇐⇒ |A| = 0 ⇐⇒ Ax = 0 has anonzero solution ⇐⇒ Columns of A are linearly dependent.
Eigenvalues and Eigenvectors.
The equation
Ax = y
can be viewed as a linear transformation that maps (or transforms)a given vector x into a new vector x.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.3
A matrix A is singular ⇐⇒ |A| = 0 ⇐⇒ Ax = 0 has anonzero solution ⇐⇒ Columns of A are linearly dependent.
Eigenvalues and Eigenvectors.
The equation
Ax = y
can be viewed as a linear transformation that maps (or transforms)a given vector x into a new vector x.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.3
A matrix A is singular ⇐⇒ |A| = 0 ⇐⇒ Ax = 0 has anonzero solution ⇐⇒ Columns of A are linearly dependent.
Eigenvalues and Eigenvectors.
The equation
Ax = y
can be viewed as a linear transformation that maps (or transforms)a given vector x into a new vector x.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.3
A matrix A is singular ⇐⇒ |A| = 0 ⇐⇒ Ax = 0 has anonzero solution ⇐⇒ Columns of A are linearly dependent.
Eigenvalues and Eigenvectors.
The equation
Ax = y
can be viewed as a linear transformation that maps (or transforms)a given vector x into a new vector x.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.3
A matrix A is singular ⇐⇒ |A| = 0 ⇐⇒ Ax = 0 has anonzero solution ⇐⇒ Columns of A are linearly dependent.
Eigenvalues and Eigenvectors.
The equation
Ax = y
can be viewed as a linear transformation that maps (or transforms)a given vector x into a new vector x.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Given an n × n matrix A we consider the problem of finding avector x that is transformed into a multiple of itself
Ax = λx
but this is equivalent to say that
Ax = λIx
Ax− λIx = 0
(A− λI) x = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Given an n × n matrix A
we consider the problem of finding avector x that is transformed into a multiple of itself
Ax = λx
but this is equivalent to say that
Ax = λIx
Ax− λIx = 0
(A− λI) x = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Given an n × n matrix A we consider the problem of finding avector x
that is transformed into a multiple of itself
Ax = λx
but this is equivalent to say that
Ax = λIx
Ax− λIx = 0
(A− λI) x = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Given an n × n matrix A we consider the problem of finding avector x that is transformed into a multiple of itself
Ax = λx
but this is equivalent to say that
Ax = λIx
Ax− λIx = 0
(A− λI) x = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Given an n × n matrix A we consider the problem of finding avector x that is transformed into a multiple of itself
Ax = λx
but this is equivalent to say that
Ax = λIx
Ax− λIx = 0
(A− λI) x = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Given an n × n matrix A we consider the problem of finding avector x that is transformed into a multiple of itself
Ax = λx
but this is equivalent to say that
Ax = λIx
Ax− λIx = 0
(A− λI) x = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Given an n × n matrix A we consider the problem of finding avector x that is transformed into a multiple of itself
Ax = λx
but this is equivalent to say that
Ax = λIx
Ax− λIx = 0
(A− λI) x = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Given an n × n matrix A we consider the problem of finding avector x that is transformed into a multiple of itself
Ax = λx
but this is equivalent to say that
Ax = λIx
Ax− λIx = 0
(A− λI) x = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Given an n × n matrix A we consider the problem of finding avector x that is transformed into a multiple of itself
Ax = λx
but this is equivalent to say that
Ax = λIx
Ax− λIx = 0
(A− λI) x = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The latter equation has nonzero solutions if and only if λ is chosenso that
|A− λI| = 0
This is a polynomial equation of degree n in λ and is called thecharacteristic equation of the matrix A
Values of λ may be either real- or complex-valued and are calledeigenvalues of A . The nonzero vectors that are obtained by usingsuch a value of λ are called the eigenvectors corresponding tothat eigenvalue.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The latter equation has nonzero solutions if and only if λ is chosenso that
|A− λI| = 0
This is a polynomial equation of degree n in λ and is called thecharacteristic equation of the matrix A
Values of λ may be either real- or complex-valued and are calledeigenvalues of A . The nonzero vectors that are obtained by usingsuch a value of λ are called the eigenvectors corresponding tothat eigenvalue.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The latter equation has nonzero solutions if and only if λ is chosenso that
|A− λI| = 0
This is a polynomial equation of degree n in λ and is called thecharacteristic equation of the matrix A
Values of λ may be either real- or complex-valued and are calledeigenvalues of A . The nonzero vectors that are obtained by usingsuch a value of λ are called the eigenvectors corresponding tothat eigenvalue.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The latter equation has nonzero solutions if and only if λ is chosenso that
|A− λI| = 0
This is a polynomial equation of degree n in λ
and is called thecharacteristic equation of the matrix A
Values of λ may be either real- or complex-valued and are calledeigenvalues of A . The nonzero vectors that are obtained by usingsuch a value of λ are called the eigenvectors corresponding tothat eigenvalue.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The latter equation has nonzero solutions if and only if λ is chosenso that
|A− λI| = 0
This is a polynomial equation of degree n in λ and is called
thecharacteristic equation of the matrix A
Values of λ may be either real- or complex-valued and are calledeigenvalues of A . The nonzero vectors that are obtained by usingsuch a value of λ are called the eigenvectors corresponding tothat eigenvalue.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The latter equation has nonzero solutions if and only if λ is chosenso that
|A− λI| = 0
This is a polynomial equation of degree n in λ and is called thecharacteristic equation of the matrix A
Values of λ may be either real- or complex-valued and are calledeigenvalues of A . The nonzero vectors that are obtained by usingsuch a value of λ are called the eigenvectors corresponding tothat eigenvalue.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The latter equation has nonzero solutions if and only if λ is chosenso that
|A− λI| = 0
This is a polynomial equation of degree n in λ and is called thecharacteristic equation of the matrix A
Values of λ may be either real- or complex-valued and are calledeigenvalues of A .
The nonzero vectors that are obtained by usingsuch a value of λ are called the eigenvectors corresponding tothat eigenvalue.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The latter equation has nonzero solutions if and only if λ is chosenso that
|A− λI| = 0
This is a polynomial equation of degree n in λ and is called thecharacteristic equation of the matrix A
Values of λ may be either real- or complex-valued and are calledeigenvalues of A . The nonzero vectors that are obtained
by usingsuch a value of λ are called the eigenvectors corresponding tothat eigenvalue.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The latter equation has nonzero solutions if and only if λ is chosenso that
|A− λI| = 0
This is a polynomial equation of degree n in λ and is called thecharacteristic equation of the matrix A
Values of λ may be either real- or complex-valued and are calledeigenvalues of A . The nonzero vectors that are obtained by usingsuch a value of λ are called
the eigenvectors corresponding tothat eigenvalue.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The latter equation has nonzero solutions if and only if λ is chosenso that
|A− λI| = 0
This is a polynomial equation of degree n in λ and is called thecharacteristic equation of the matrix A
Values of λ may be either real- or complex-valued and are calledeigenvalues of A . The nonzero vectors that are obtained by usingsuch a value of λ are called the eigenvectors corresponding tothat eigenvalue.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Facts
a) It is possible to show that if λ1 and λ2 are two eigenvalues of Aandif λ1 6= λ2, then their corresponding eigenvectors x (1) and x (2)
are linearly independent.
This result extends to any set λ1, ..., λk of distinct eigenvalues:their eigenvectors x (1), ..., x (k) are linearly independent. Thus, ifeach eigenvalue of an n×n matrix is simple,then the n eigenvectorsof A , one for each eigenvalue, are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Facts
a) It is possible to show that if λ1 and λ2 are two eigenvalues of Aandif λ1 6= λ2, then their corresponding eigenvectors x (1) and x (2)
are linearly independent.
This result extends to any set λ1, ..., λk of distinct eigenvalues:their eigenvectors x (1), ..., x (k) are linearly independent. Thus, ifeach eigenvalue of an n×n matrix is simple,then the n eigenvectorsof A , one for each eigenvalue, are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Facts
a) It is possible to show that if λ1 and λ2
are two eigenvalues of Aandif λ1 6= λ2, then their corresponding eigenvectors x (1) and x (2)
are linearly independent.
This result extends to any set λ1, ..., λk of distinct eigenvalues:their eigenvectors x (1), ..., x (k) are linearly independent. Thus, ifeach eigenvalue of an n×n matrix is simple,then the n eigenvectorsof A , one for each eigenvalue, are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Facts
a) It is possible to show that if λ1 and λ2 are two eigenvalues of Aand
if λ1 6= λ2, then their corresponding eigenvectors x (1) and x (2)
are linearly independent.
This result extends to any set λ1, ..., λk of distinct eigenvalues:their eigenvectors x (1), ..., x (k) are linearly independent. Thus, ifeach eigenvalue of an n×n matrix is simple,then the n eigenvectorsof A , one for each eigenvalue, are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Facts
a) It is possible to show that if λ1 and λ2 are two eigenvalues of Aandif λ1 6= λ2, then
their corresponding eigenvectors x (1) and x (2)
are linearly independent.
This result extends to any set λ1, ..., λk of distinct eigenvalues:their eigenvectors x (1), ..., x (k) are linearly independent. Thus, ifeach eigenvalue of an n×n matrix is simple,then the n eigenvectorsof A , one for each eigenvalue, are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Facts
a) It is possible to show that if λ1 and λ2 are two eigenvalues of Aandif λ1 6= λ2, then their corresponding eigenvectors x (1) and x (2)
are linearly independent.
This result extends to any set λ1, ..., λk of distinct eigenvalues:their eigenvectors x (1), ..., x (k) are linearly independent. Thus, ifeach eigenvalue of an n×n matrix is simple,then the n eigenvectorsof A , one for each eigenvalue, are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Facts
a) It is possible to show that if λ1 and λ2 are two eigenvalues of Aandif λ1 6= λ2, then their corresponding eigenvectors x (1) and x (2)
are linearly independent.
This result extends to any set λ1, ..., λk of distinct eigenvalues:
their eigenvectors x (1), ..., x (k) are linearly independent. Thus, ifeach eigenvalue of an n×n matrix is simple,then the n eigenvectorsof A , one for each eigenvalue, are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Facts
a) It is possible to show that if λ1 and λ2 are two eigenvalues of Aandif λ1 6= λ2, then their corresponding eigenvectors x (1) and x (2)
are linearly independent.
This result extends to any set λ1, ..., λk of distinct eigenvalues:their eigenvectors x (1), ..., x (k) are linearly independent.
Thus, ifeach eigenvalue of an n×n matrix is simple,then the n eigenvectorsof A , one for each eigenvalue, are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Facts
a) It is possible to show that if λ1 and λ2 are two eigenvalues of Aandif λ1 6= λ2, then their corresponding eigenvectors x (1) and x (2)
are linearly independent.
This result extends to any set λ1, ..., λk of distinct eigenvalues:their eigenvectors x (1), ..., x (k) are linearly independent. Thus, ifeach eigenvalue of an n×n matrix is simple,
then the n eigenvectorsof A , one for each eigenvalue, are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Facts
a) It is possible to show that if λ1 and λ2 are two eigenvalues of Aandif λ1 6= λ2, then their corresponding eigenvectors x (1) and x (2)
are linearly independent.
This result extends to any set λ1, ..., λk of distinct eigenvalues:their eigenvectors x (1), ..., x (k) are linearly independent. Thus, ifeach eigenvalue of an n×n matrix is simple,then the n eigenvectorsof A ,
one for each eigenvalue, are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Facts
a) It is possible to show that if λ1 and λ2 are two eigenvalues of Aandif λ1 6= λ2, then their corresponding eigenvectors x (1) and x (2)
are linearly independent.
This result extends to any set λ1, ..., λk of distinct eigenvalues:their eigenvectors x (1), ..., x (k) are linearly independent. Thus, ifeach eigenvalue of an n×n matrix is simple,then the n eigenvectorsof A , one for each eigenvalue, are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
b) On the other hand, if A has one or more repeated eigenvalues,then there may be fewer than n linearly independent eigenvectorsassociated with A, since for a repeated eigenvalue with multiplicitym, we may have q < m linearly independent vectors.
c) In the case of an eigenvalue, λi with multiplicity m, if we canfind m eigenvectors x (i1), x (i2), ..., x (im), linearly independentassociated to λi , we say that the matrix is Non-defective.
d) Otherwise, if we are able to find just x (i1), x (i2), ..., x (iq);q < m linearly independent associated to λi , we say that thematrix is Defective.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
b) On the other hand,
if A has one or more repeated eigenvalues,then there may be fewer than n linearly independent eigenvectorsassociated with A, since for a repeated eigenvalue with multiplicitym, we may have q < m linearly independent vectors.
c) In the case of an eigenvalue, λi with multiplicity m, if we canfind m eigenvectors x (i1), x (i2), ..., x (im), linearly independentassociated to λi , we say that the matrix is Non-defective.
d) Otherwise, if we are able to find just x (i1), x (i2), ..., x (iq);q < m linearly independent associated to λi , we say that thematrix is Defective.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
b) On the other hand, if A has one or more repeated eigenvalues,
then there may be fewer than n linearly independent eigenvectorsassociated with A, since for a repeated eigenvalue with multiplicitym, we may have q < m linearly independent vectors.
c) In the case of an eigenvalue, λi with multiplicity m, if we canfind m eigenvectors x (i1), x (i2), ..., x (im), linearly independentassociated to λi , we say that the matrix is Non-defective.
d) Otherwise, if we are able to find just x (i1), x (i2), ..., x (iq);q < m linearly independent associated to λi , we say that thematrix is Defective.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
b) On the other hand, if A has one or more repeated eigenvalues,then there may be fewer than n linearly independent eigenvectorsassociated with A,
since for a repeated eigenvalue with multiplicitym, we may have q < m linearly independent vectors.
c) In the case of an eigenvalue, λi with multiplicity m, if we canfind m eigenvectors x (i1), x (i2), ..., x (im), linearly independentassociated to λi , we say that the matrix is Non-defective.
d) Otherwise, if we are able to find just x (i1), x (i2), ..., x (iq);q < m linearly independent associated to λi , we say that thematrix is Defective.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
b) On the other hand, if A has one or more repeated eigenvalues,then there may be fewer than n linearly independent eigenvectorsassociated with A, since for a repeated eigenvalue with multiplicitym, we may have q < m linearly independent vectors.
c) In the case of an eigenvalue, λi with multiplicity m, if we canfind m eigenvectors x (i1), x (i2), ..., x (im), linearly independentassociated to λi , we say that the matrix is Non-defective.
d) Otherwise, if we are able to find just x (i1), x (i2), ..., x (iq);q < m linearly independent associated to λi , we say that thematrix is Defective.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
b) On the other hand, if A has one or more repeated eigenvalues,then there may be fewer than n linearly independent eigenvectorsassociated with A, since for a repeated eigenvalue with multiplicitym, we may have q < m linearly independent vectors.
c) In the case of an eigenvalue, λi with multiplicity m, if
we canfind m eigenvectors x (i1), x (i2), ..., x (im), linearly independentassociated to λi , we say that the matrix is Non-defective.
d) Otherwise, if we are able to find just x (i1), x (i2), ..., x (iq);q < m linearly independent associated to λi , we say that thematrix is Defective.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
b) On the other hand, if A has one or more repeated eigenvalues,then there may be fewer than n linearly independent eigenvectorsassociated with A, since for a repeated eigenvalue with multiplicitym, we may have q < m linearly independent vectors.
c) In the case of an eigenvalue, λi with multiplicity m, if we canfind m eigenvectors x (i1), x (i2), ..., x (im),
linearly independentassociated to λi , we say that the matrix is Non-defective.
d) Otherwise, if we are able to find just x (i1), x (i2), ..., x (iq);q < m linearly independent associated to λi , we say that thematrix is Defective.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
b) On the other hand, if A has one or more repeated eigenvalues,then there may be fewer than n linearly independent eigenvectorsassociated with A, since for a repeated eigenvalue with multiplicitym, we may have q < m linearly independent vectors.
c) In the case of an eigenvalue, λi with multiplicity m, if we canfind m eigenvectors x (i1), x (i2), ..., x (im), linearly independentassociated to λi , we say that the matrix is Non-defective.
d) Otherwise, if we are able to find just x (i1), x (i2), ..., x (iq);q < m linearly independent associated to λi , we say that thematrix is Defective.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
b) On the other hand, if A has one or more repeated eigenvalues,then there may be fewer than n linearly independent eigenvectorsassociated with A, since for a repeated eigenvalue with multiplicitym, we may have q < m linearly independent vectors.
c) In the case of an eigenvalue, λi with multiplicity m, if we canfind m eigenvectors x (i1), x (i2), ..., x (im), linearly independentassociated to λi , we say that the matrix is Non-defective.
d) Otherwise, if we are able to find
just x (i1), x (i2), ..., x (iq);q < m linearly independent associated to λi , we say that thematrix is Defective.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
b) On the other hand, if A has one or more repeated eigenvalues,then there may be fewer than n linearly independent eigenvectorsassociated with A, since for a repeated eigenvalue with multiplicitym, we may have q < m linearly independent vectors.
c) In the case of an eigenvalue, λi with multiplicity m, if we canfind m eigenvectors x (i1), x (i2), ..., x (im), linearly independentassociated to λi , we say that the matrix is Non-defective.
d) Otherwise, if we are able to find just x (i1), x (i2), ..., x (iq);q < m
linearly independent associated to λi , we say that thematrix is Defective.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
b) On the other hand, if A has one or more repeated eigenvalues,then there may be fewer than n linearly independent eigenvectorsassociated with A, since for a repeated eigenvalue with multiplicitym, we may have q < m linearly independent vectors.
c) In the case of an eigenvalue, λi with multiplicity m, if we canfind m eigenvectors x (i1), x (i2), ..., x (im), linearly independentassociated to λi , we say that the matrix is Non-defective.
d) Otherwise, if we are able to find just x (i1), x (i2), ..., x (iq);q < m linearly independent associated to λi , we say that thematrix is Defective.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.10
Find the eigenvalues and eigenvectors of the matrix
A =
0 1 11 0 11 1 0
Solution
The eigenvalues λ and eigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.10
Find the eigenvalues and eigenvectors of the matrix
A =
0 1 11 0 11 1 0
Solution
The eigenvalues λ and eigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.10
Find the eigenvalues and eigenvectors of the matrix
A =
0 1 11 0 11 1 0
Solution
The eigenvalues λ and eigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.10
Find the eigenvalues and eigenvectors of the matrix
A =
0 1 11 0 11 1 0
Solution
The eigenvalues λ and eigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.10
Find the eigenvalues and eigenvectors of the matrix
A =
0 1 11 0 11 1 0
Solution
The eigenvalues λ and eigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.10
Find the eigenvalues and eigenvectors of the matrix
A =
0 1 11 0 11 1 0
Solution
The eigenvalues λ and
eigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.10
Find the eigenvalues and eigenvectors of the matrix
A =
0 1 11 0 11 1 0
Solution
The eigenvalues λ and eigenvectors x satisfy the equation
(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.10
Find the eigenvalues and eigenvectors of the matrix
A =
0 1 11 0 11 1 0
Solution
The eigenvalues λ and eigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
−λ 1 11 −λ 11 1 −λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣−λ 1 11 −λ 11 1 −λ
∣∣∣∣∣∣ = −
∣∣∣∣∣∣1 −λ 1−λ 1 11 1 −λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 −λ 11 1 −λ−λ 1 1
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 −λ 10 λ+ 1 −1− λ0 −λ2 + 1 λ+ 1
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 0 0−λ λ+ 1 −λ2 + 11 −1− λ λ+ 1
∣∣∣∣∣∣ = − λ3 + 3λ2 + 2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
−λ 1 11 −λ 11 1 −λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣−λ 1 11 −λ 11 1 −λ
∣∣∣∣∣∣ = −
∣∣∣∣∣∣1 −λ 1−λ 1 11 1 −λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 −λ 11 1 −λ−λ 1 1
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 −λ 10 λ+ 1 −1− λ0 −λ2 + 1 λ+ 1
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 0 0−λ λ+ 1 −λ2 + 11 −1− λ λ+ 1
∣∣∣∣∣∣ = − λ3 + 3λ2 + 2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
−λ 1 11 −λ 11 1 −λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣−λ 1 11 −λ 11 1 −λ
∣∣∣∣∣∣ = −
∣∣∣∣∣∣1 −λ 1−λ 1 11 1 −λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 −λ 11 1 −λ−λ 1 1
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 −λ 10 λ+ 1 −1− λ0 −λ2 + 1 λ+ 1
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 0 0−λ λ+ 1 −λ2 + 11 −1− λ λ+ 1
∣∣∣∣∣∣ = − λ3 + 3λ2 + 2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
−λ 1 11 −λ 11 1 −λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣−λ 1 11 −λ 11 1 −λ
∣∣∣∣∣∣ =
−
∣∣∣∣∣∣1 −λ 1−λ 1 11 1 −λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 −λ 11 1 −λ−λ 1 1
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 −λ 10 λ+ 1 −1− λ0 −λ2 + 1 λ+ 1
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 0 0−λ λ+ 1 −λ2 + 11 −1− λ λ+ 1
∣∣∣∣∣∣ = − λ3 + 3λ2 + 2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
−λ 1 11 −λ 11 1 −λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣−λ 1 11 −λ 11 1 −λ
∣∣∣∣∣∣ = −
∣∣∣∣∣∣1 −λ 1−λ 1 11 1 −λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 −λ 11 1 −λ−λ 1 1
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 −λ 10 λ+ 1 −1− λ0 −λ2 + 1 λ+ 1
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 0 0−λ λ+ 1 −λ2 + 11 −1− λ λ+ 1
∣∣∣∣∣∣ = − λ3 + 3λ2 + 2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
−λ 1 11 −λ 11 1 −λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣−λ 1 11 −λ 11 1 −λ
∣∣∣∣∣∣ = −
∣∣∣∣∣∣1 −λ 1−λ 1 11 1 −λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 −λ 11 1 −λ−λ 1 1
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 −λ 10 λ+ 1 −1− λ0 −λ2 + 1 λ+ 1
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 0 0−λ λ+ 1 −λ2 + 11 −1− λ λ+ 1
∣∣∣∣∣∣ = − λ3 + 3λ2 + 2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
−λ 1 11 −λ 11 1 −λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣−λ 1 11 −λ 11 1 −λ
∣∣∣∣∣∣ = −
∣∣∣∣∣∣1 −λ 1−λ 1 11 1 −λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 −λ 11 1 −λ−λ 1 1
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 −λ 10 λ+ 1 −1− λ0 −λ2 + 1 λ+ 1
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 0 0−λ λ+ 1 −λ2 + 11 −1− λ λ+ 1
∣∣∣∣∣∣ =
− λ3 + 3λ2 + 2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
−λ 1 11 −λ 11 1 −λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣−λ 1 11 −λ 11 1 −λ
∣∣∣∣∣∣ = −
∣∣∣∣∣∣1 −λ 1−λ 1 11 1 −λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 −λ 11 1 −λ−λ 1 1
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 −λ 10 λ+ 1 −1− λ0 −λ2 + 1 λ+ 1
∣∣∣∣∣∣ =
∣∣∣∣∣∣1 0 0−λ λ+ 1 −λ2 + 11 −1− λ λ+ 1
∣∣∣∣∣∣ = − λ3 + 3λ2 + 2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 2, λ2 = −1, and λ3 = −1 .
1) For λ1 = 2
−λ1 1 11 −λ1 11 1 −λ1
x1x2x3
=
−2 1 11 −2 11 1 −2
x1x2x3
=
000
We can reduce this to the equivalent system2 −1 −1
0 1 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 2, λ2 = −1, and λ3 = −1 .
1) For λ1 = 2
−λ1 1 11 −λ1 11 1 −λ1
x1x2x3
=
−2 1 11 −2 11 1 −2
x1x2x3
=
000
We can reduce this to the equivalent system2 −1 −1
0 1 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 2, λ2 = −1, and λ3 = −1 .
1) For λ1 = 2
−λ1 1 11 −λ1 11 1 −λ1
x1x2x3
=
−2 1 11 −2 11 1 −2
x1x2x3
=
000
We can reduce this to the equivalent system2 −1 −1
0 1 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 2, λ2 = −1, and λ3 = −1 .
1) For λ1 = 2
−λ1 1 11 −λ1 11 1 −λ1
x1x2x3
=
−2 1 11 −2 11 1 −2
x1x2x3
=
000
We can reduce this to the equivalent system2 −1 −1
0 1 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 2, λ2 = −1, and λ3 = −1 .
1) For λ1 = 2
−λ1 1 11 −λ1 11 1 −λ1
x1x2x3
=
−2 1 11 −2 11 1 −2
x1x2x3
=
000
We can reduce this to the equivalent system2 −1 −1
0 1 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 2, λ2 = −1, and λ3 = −1 .
1) For λ1 = 2
−λ1 1 11 −λ1 11 1 −λ1
x1x2x3
=
−2 1 11 −2 11 1 −2
x1x2x3
=
000
We can reduce this to the equivalent system2 −1 −10 1 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 2, λ2 = −1, and λ3 = −1 .
1) For λ1 = 2
−λ1 1 11 −λ1 11 1 −λ1
x1x2x3
=
−2 1 11 −2 11 1 −2
x1x2x3
=
000
We can reduce this to the equivalent system
2 −1 −10 1 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 2, λ2 = −1, and λ3 = −1 .
1) For λ1 = 2
−λ1 1 11 −λ1 11 1 −λ1
x1x2x3
=
−2 1 11 −2 11 1 −2
x1x2x3
=
000
We can reduce this to the equivalent system2 −1 −1
0 1 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
2x1 − x2 − x3 = 0 x2 − x3 = 0
Two equations and three unknowns. Hence, one of them is a freevariable, let’s say x3 = α. Therefore x2 = x3 = α andx1 = (x2 + x3)/2 = α. Thus we have
x =
ααα
= α
111
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
2x1 − x2 − x3 = 0 x2 − x3 = 0
Two equations and three unknowns. Hence, one of them is a freevariable, let’s say x3 = α. Therefore x2 = x3 = α andx1 = (x2 + x3)/2 = α. Thus we have
x =
ααα
= α
111
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
2x1 − x2 − x3 = 0 x2 − x3 = 0
Two equations and three unknowns. Hence, one of them is a freevariable, let’s say x3 = α. Therefore x2 = x3 = α andx1 = (x2 + x3)/2 = α. Thus we have
x =
ααα
= α
111
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
2x1 − x2 − x3 = 0 x2 − x3 = 0
Two equations and three unknowns. Hence,
one of them is a freevariable, let’s say x3 = α. Therefore x2 = x3 = α andx1 = (x2 + x3)/2 = α. Thus we have
x =
ααα
= α
111
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
2x1 − x2 − x3 = 0 x2 − x3 = 0
Two equations and three unknowns. Hence, one of them is a freevariable,
let’s say x3 = α. Therefore x2 = x3 = α andx1 = (x2 + x3)/2 = α. Thus we have
x =
ααα
= α
111
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
2x1 − x2 − x3 = 0 x2 − x3 = 0
Two equations and three unknowns. Hence, one of them is a freevariable, let’s say x3 = α. Therefore
x2 = x3 = α andx1 = (x2 + x3)/2 = α. Thus we have
x =
ααα
= α
111
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
2x1 − x2 − x3 = 0 x2 − x3 = 0
Two equations and three unknowns. Hence, one of them is a freevariable, let’s say x3 = α. Therefore x2 = x3 = α andx1 = (x2 + x3)/2 = α.
Thus we have
x =
ααα
= α
111
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
2x1 − x2 − x3 = 0 x2 − x3 = 0
Two equations and three unknowns. Hence, one of them is a freevariable, let’s say x3 = α. Therefore x2 = x3 = α andx1 = (x2 + x3)/2 = α. Thus we have
x =
ααα
= α
111
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
2x1 − x2 − x3 = 0 x2 − x3 = 0
Two equations and three unknowns. Hence, one of them is a freevariable, let’s say x3 = α. Therefore x2 = x3 = α andx1 = (x2 + x3)/2 = α. Thus we have
x =
ααα
= α
111
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In particular, we have the eigenvector
x(1) =
111
2) For λ2 = −1
−λ2 1 11 −λ2 11 1 −λ2
x1x2x3
=
1 1 11 1 11 1 1
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In particular, we have the eigenvector
x(1) =
111
2) For λ2 = −1
−λ2 1 11 −λ2 11 1 −λ2
x1x2x3
=
1 1 11 1 11 1 1
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In particular, we have the eigenvector
x(1) =
111
2) For λ2 = −1
−λ2 1 11 −λ2 11 1 −λ2
x1x2x3
=
1 1 11 1 11 1 1
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In particular, we have the eigenvector
x(1) =
111
2) For λ2 = −1
−λ2 1 11 −λ2 11 1 −λ2
x1x2x3
=
1 1 11 1 11 1 1
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In particular, we have the eigenvector
x(1) =
111
2) For λ2 = −1
−λ2 1 11 −λ2 11 1 −λ2
x1x2x3
=
1 1 11 1 11 1 1
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In particular, we have the eigenvector
x(1) =
111
2) For λ2 = −1
−λ2 1 11 −λ2 11 1 −λ2
x1x2x3
=
1 1 11 1 11 1 1
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the single equation
x1 + x2 + x3 = 0
One equation and three unknowns. Hence, two of them are freeveriables, let’s say x1 = α, x2 = β, and x3 = −α− β . Thus wehave
x =
αβ
−α− β
= α
10−1
+ β
01−1
; α, β = real numbers
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the single equation
x1 + x2 + x3 = 0
One equation and three unknowns. Hence, two of them are freeveriables, let’s say x1 = α, x2 = β, and x3 = −α− β . Thus wehave
x =
αβ
−α− β
= α
10−1
+ β
01−1
; α, β = real numbers
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the single equation
x1 + x2 + x3 = 0
One equation and three unknowns. Hence, two of them are freeveriables, let’s say x1 = α, x2 = β, and x3 = −α− β . Thus wehave
x =
αβ
−α− β
= α
10−1
+ β
01−1
; α, β = real numbers
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the single equation
x1 + x2 + x3 = 0
One equation and three unknowns. Hence,
two of them are freeveriables, let’s say x1 = α, x2 = β, and x3 = −α− β . Thus wehave
x =
αβ
−α− β
= α
10−1
+ β
01−1
; α, β = real numbers
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the single equation
x1 + x2 + x3 = 0
One equation and three unknowns. Hence, two of them are freeveriables,
let’s say x1 = α, x2 = β, and x3 = −α− β . Thus wehave
x =
αβ
−α− β
= α
10−1
+ β
01−1
; α, β = real numbers
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the single equation
x1 + x2 + x3 = 0
One equation and three unknowns. Hence, two of them are freeveriables, let’s say x1 = α,
x2 = β, and x3 = −α− β . Thus wehave
x =
αβ
−α− β
= α
10−1
+ β
01−1
; α, β = real numbers
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the single equation
x1 + x2 + x3 = 0
One equation and three unknowns. Hence, two of them are freeveriables, let’s say x1 = α, x2 = β, and
x3 = −α− β . Thus wehave
x =
αβ
−α− β
= α
10−1
+ β
01−1
; α, β = real numbers
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the single equation
x1 + x2 + x3 = 0
One equation and three unknowns. Hence, two of them are freeveriables, let’s say x1 = α, x2 = β, and x3 = −α− β .
Thus wehave
x =
αβ
−α− β
= α
10−1
+ β
01−1
; α, β = real numbers
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the single equation
x1 + x2 + x3 = 0
One equation and three unknowns. Hence, two of them are freeveriables, let’s say x1 = α, x2 = β, and x3 = −α− β . Thus wehave
x =
αβ
−α− β
= α
10−1
+ β
01−1
; α, β = real numbers
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the single equation
x1 + x2 + x3 = 0
One equation and three unknowns. Hence, two of them are freeveriables, let’s say x1 = α, x2 = β, and x3 = −α− β . Thus wehave
x =
αβ
−α− β
= α
10−1
+ β
01−1
; α, β = real numbers
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way two linearly independent eigenvectors associated toλ2 = −1 are ( α = β = 1 )
x(2) =
10
− 1
x(3) =
01
− 1
Thus, the three linearly independent eigenvectors, are
x(1) =
111
x(2) =
10
− 1
x(3) =
01
− 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way two linearly independent eigenvectors associated toλ2 = −1 are ( α = β = 1 )
x(2) =
10
− 1
x(3) =
01
− 1
Thus, the three linearly independent eigenvectors, are
x(1) =
111
x(2) =
10
− 1
x(3) =
01
− 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way two linearly independent eigenvectors associated toλ2 = −1 are ( α = β = 1 )
x(2) =
10
− 1
x(3) =
01
− 1
Thus, the three linearly independent eigenvectors, are
x(1) =
111
x(2) =
10
− 1
x(3) =
01
− 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way two linearly independent eigenvectors associated toλ2 = −1 are ( α = β = 1 )
x(2) =
10
− 1
x(3) =
01
− 1
Thus, the three linearly independent eigenvectors, are
x(1) =
111
x(2) =
10
− 1
x(3) =
01
− 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way two linearly independent eigenvectors associated toλ2 = −1 are ( α = β = 1 )
x(2) =
10
− 1
x(3) =
01
− 1
Thus, the three linearly independent eigenvectors, are
x(1) =
111
x(2) =
10
− 1
x(3) =
01
− 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way two linearly independent eigenvectors associated toλ2 = −1 are ( α = β = 1 )
x(2) =
10
− 1
x(3) =
01
− 1
Thus, the three linearly independent eigenvectors, are
x(1) =
111
x(2) =
10
− 1
x(3) =
01
− 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way two linearly independent eigenvectors associated toλ2 = −1 are ( α = β = 1 )
x(2) =
10
− 1
x(3) =
01
− 1
Thus, the three linearly independent eigenvectors, are
x(1) =
111
x(2) =
10
− 1
x(3) =
01
− 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way two linearly independent eigenvectors associated toλ2 = −1 are ( α = β = 1 )
x(2) =
10
− 1
x(3) =
01
− 1
Thus, the three linearly independent eigenvectors, are
x(1) =
111
x(2) =
10
− 1
x(3) =
01
− 1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.11
Find the eigenvalues and eigenvectors of the matrix
A =
2 −3 −10 −1 0−1 1 2
Solution
The eigenvalues λ andeigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.11
Find the eigenvalues and eigenvectors of the matrix
A =
2 −3 −10 −1 0−1 1 2
Solution
The eigenvalues λ andeigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.11
Find the eigenvalues and eigenvectors of the matrix
A =
2 −3 −10 −1 0−1 1 2
Solution
The eigenvalues λ andeigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.11
Find the eigenvalues and eigenvectors of the matrix
A =
2 −3 −10 −1 0−1 1 2
Solution
The eigenvalues λ andeigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.11
Find the eigenvalues and eigenvectors of the matrix
A =
2 −3 −10 −1 0−1 1 2
Solution
The eigenvalues λ andeigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.11
Find the eigenvalues and eigenvectors of the matrix
A =
2 −3 −10 −1 0−1 1 2
Solution
The eigenvalues λ and
eigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.11
Find the eigenvalues and eigenvectors of the matrix
A =
2 −3 −10 −1 0−1 1 2
Solution
The eigenvalues λ andeigenvectors x satisfy the equation
(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.11
Find the eigenvalues and eigenvectors of the matrix
A =
2 −3 −10 −1 0−1 1 2
Solution
The eigenvalues λ andeigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣2− λ −3 −1
0 −1− λ 0−1 1 2− λ
∣∣∣∣∣∣ = −
∣∣∣∣∣∣2− λ −3 −1−1 1 2− λ0 −1− λ 0
∣∣∣∣∣∣ =
∣∣∣∣∣∣−1 1 2− λ
2− λ −3 −10 −1− λ 0
∣∣∣∣∣∣ =
∣∣∣∣∣∣−1 1 2− λ0 −1− λ (2− λ)2 − 10 −1− λ 0
∣∣∣∣∣∣ =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣2− λ −3 −1
0 −1− λ 0−1 1 2− λ
∣∣∣∣∣∣ = −
∣∣∣∣∣∣2− λ −3 −1−1 1 2− λ0 −1− λ 0
∣∣∣∣∣∣ =
∣∣∣∣∣∣−1 1 2− λ
2− λ −3 −10 −1− λ 0
∣∣∣∣∣∣ =
∣∣∣∣∣∣−1 1 2− λ0 −1− λ (2− λ)2 − 10 −1− λ 0
∣∣∣∣∣∣ =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣2− λ −3 −1
0 −1− λ 0−1 1 2− λ
∣∣∣∣∣∣ = −
∣∣∣∣∣∣2− λ −3 −1−1 1 2− λ0 −1− λ 0
∣∣∣∣∣∣ =
∣∣∣∣∣∣−1 1 2− λ
2− λ −3 −10 −1− λ 0
∣∣∣∣∣∣ =
∣∣∣∣∣∣−1 1 2− λ0 −1− λ (2− λ)2 − 10 −1− λ 0
∣∣∣∣∣∣ =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣2− λ −3 −1
0 −1− λ 0−1 1 2− λ
∣∣∣∣∣∣ = −
∣∣∣∣∣∣2− λ −3 −1−1 1 2− λ0 −1− λ 0
∣∣∣∣∣∣ =
∣∣∣∣∣∣−1 1 2− λ
2− λ −3 −10 −1− λ 0
∣∣∣∣∣∣ =
∣∣∣∣∣∣−1 1 2− λ0 −1− λ (2− λ)2 − 10 −1− λ 0
∣∣∣∣∣∣ =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣2− λ −3 −1
0 −1− λ 0−1 1 2− λ
∣∣∣∣∣∣ =
−
∣∣∣∣∣∣2− λ −3 −1−1 1 2− λ0 −1− λ 0
∣∣∣∣∣∣ =
∣∣∣∣∣∣−1 1 2− λ
2− λ −3 −10 −1− λ 0
∣∣∣∣∣∣ =
∣∣∣∣∣∣−1 1 2− λ0 −1− λ (2− λ)2 − 10 −1− λ 0
∣∣∣∣∣∣ =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣2− λ −3 −1
0 −1− λ 0−1 1 2− λ
∣∣∣∣∣∣ = −
∣∣∣∣∣∣2− λ −3 −1−1 1 2− λ0 −1− λ 0
∣∣∣∣∣∣ =
∣∣∣∣∣∣−1 1 2− λ
2− λ −3 −10 −1− λ 0
∣∣∣∣∣∣ =
∣∣∣∣∣∣−1 1 2− λ0 −1− λ (2− λ)2 − 10 −1− λ 0
∣∣∣∣∣∣ =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣2− λ −3 −1
0 −1− λ 0−1 1 2− λ
∣∣∣∣∣∣ = −
∣∣∣∣∣∣2− λ −3 −1−1 1 2− λ0 −1− λ 0
∣∣∣∣∣∣ =
∣∣∣∣∣∣−1 1 2− λ
2− λ −3 −10 −1− λ 0
∣∣∣∣∣∣ =
∣∣∣∣∣∣−1 1 2− λ0 −1− λ (2− λ)2 − 10 −1− λ 0
∣∣∣∣∣∣ =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣2− λ −3 −1
0 −1− λ 0−1 1 2− λ
∣∣∣∣∣∣ = −
∣∣∣∣∣∣2− λ −3 −1−1 1 2− λ0 −1− λ 0
∣∣∣∣∣∣ =
∣∣∣∣∣∣−1 1 2− λ
2− λ −3 −10 −1− λ 0
∣∣∣∣∣∣ =
∣∣∣∣∣∣−1 1 2− λ0 −1− λ (2− λ)2 − 10 −1− λ 0
∣∣∣∣∣∣ =
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
−
∣∣∣∣∣∣−1 2− λ 10 (2− λ)2 − 1 −1− λ0 0 −1− λ
∣∣∣∣∣∣ = (1 + λ)[(2− λ)2 − 1
]= 0
The roots of are λ1 = −1, λ2 = 1, and λ3 = 3 .
1) For λ1 = −1
(A− λ1I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
−
∣∣∣∣∣∣−1 2− λ 10 (2− λ)2 − 1 −1− λ0 0 −1− λ
∣∣∣∣∣∣ =
(1 + λ)[(2− λ)2 − 1
]= 0
The roots of are λ1 = −1, λ2 = 1, and λ3 = 3 .
1) For λ1 = −1
(A− λ1I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
−
∣∣∣∣∣∣−1 2− λ 10 (2− λ)2 − 1 −1− λ0 0 −1− λ
∣∣∣∣∣∣ = (1 + λ)[(2− λ)2 − 1
]= 0
The roots of are λ1 = −1, λ2 = 1, and λ3 = 3 .
1) For λ1 = −1
(A− λ1I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
−
∣∣∣∣∣∣−1 2− λ 10 (2− λ)2 − 1 −1− λ0 0 −1− λ
∣∣∣∣∣∣ = (1 + λ)[(2− λ)2 − 1
]= 0
The roots of are λ1 = −1, λ2 = 1, and λ3 = 3 .
1) For λ1 = −1
(A− λ1I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
−
∣∣∣∣∣∣−1 2− λ 10 (2− λ)2 − 1 −1− λ0 0 −1− λ
∣∣∣∣∣∣ = (1 + λ)[(2− λ)2 − 1
]= 0
The roots of are λ1 = −1, λ2 = 1, and λ3 = 3 .
1) For λ1 = −1
(A− λ1I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
−
∣∣∣∣∣∣−1 2− λ 10 (2− λ)2 − 1 −1− λ0 0 −1− λ
∣∣∣∣∣∣ = (1 + λ)[(2− λ)2 − 1
]= 0
The roots of are λ1 = −1, λ2 = 1, and λ3 = 3 .
1) For λ1 = −1
(A− λ1I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
−
∣∣∣∣∣∣−1 2− λ 10 (2− λ)2 − 1 −1− λ0 0 −1− λ
∣∣∣∣∣∣ = (1 + λ)[(2− λ)2 − 1
]= 0
The roots of are λ1 = −1, λ2 = 1, and λ3 = 3 .
1) For λ1 = −1
(A− λ1I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
−
∣∣∣∣∣∣−1 2− λ 10 (2− λ)2 − 1 −1− λ0 0 −1− λ
∣∣∣∣∣∣ = (1 + λ)[(2− λ)2 − 1
]= 0
The roots of are λ1 = −1, λ2 = 1, and λ3 = 3 .
1) For λ1 = −1
(A− λ1I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
3 −3 −10 0 0−1 1 3
x1x2x3
=
000
We can reduce this to the equivalent system
3 −3 −10 0 01 1 3
=
1 1 30 0 03 −3 −1
=
1 1 30 0 80 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
3 −3 −10 0 0−1 1 3
x1x2x3
=
000
We can reduce this to the equivalent system
3 −3 −10 0 01 1 3
=
1 1 30 0 03 −3 −1
=
1 1 30 0 80 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
3 −3 −10 0 0−1 1 3
x1x2x3
=
000
We can reduce this to the equivalent system
3 −3 −10 0 01 1 3
=
1 1 30 0 03 −3 −1
=
1 1 30 0 80 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
3 −3 −10 0 0−1 1 3
x1x2x3
=
000
We can reduce this to the equivalent system
3 −3 −10 0 01 1 3
=
1 1 30 0 03 −3 −1
=
1 1 30 0 80 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
3 −3 −10 0 0−1 1 3
x1x2x3
=
000
We can reduce this to the equivalent system
3 −3 −10 0 01 1 3
=
1 1 30 0 03 −3 −1
=
1 1 30 0 80 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
3 −3 −10 0 0−1 1 3
x1x2x3
=
000
We can reduce this to the equivalent system
3 −3 −10 0 01 1 3
=
1 1 30 0 03 −3 −1
=
1 1 30 0 80 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 3x3 = 0 8x3 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x2 = α. Therefore we have x1 = −x2 = −α, andx3 = 0 . Thus, we get
x =
−αα0
= α
1−10
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 3x3 = 0 8x3 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x2 = α. Therefore we have x1 = −x2 = −α, andx3 = 0 . Thus, we get
x =
−αα0
= α
1−10
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 3x3 = 0 8x3 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x2 = α. Therefore we have x1 = −x2 = −α, andx3 = 0 . Thus, we get
x =
−αα0
= α
1−10
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 3x3 = 0 8x3 = 0
One equation and two unknowns. Hence,
one of them is freevariable, let’s say x2 = α. Therefore we have x1 = −x2 = −α, andx3 = 0 . Thus, we get
x =
−αα0
= α
1−10
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 3x3 = 0 8x3 = 0
One equation and two unknowns. Hence, one of them is freevariable,
let’s say x2 = α. Therefore we have x1 = −x2 = −α, andx3 = 0 . Thus, we get
x =
−αα0
= α
1−10
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 3x3 = 0 8x3 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x2 = α.
Therefore we have x1 = −x2 = −α, andx3 = 0 . Thus, we get
x =
−αα0
= α
1−10
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 3x3 = 0 8x3 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x2 = α. Therefore we have
x1 = −x2 = −α, andx3 = 0 . Thus, we get
x =
−αα0
= α
1−10
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 3x3 = 0 8x3 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x2 = α. Therefore we have x1 = −x2 = −α, and
x3 = 0 . Thus, we get
x =
−αα0
= α
1−10
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 3x3 = 0 8x3 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x2 = α. Therefore we have x1 = −x2 = −α, andx3 = 0 .
Thus, we get
x =
−αα0
= α
1−10
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 3x3 = 0 8x3 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x2 = α. Therefore we have x1 = −x2 = −α, andx3 = 0 . Thus, we get
x =
−αα0
= α
1−10
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 3x3 = 0 8x3 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x2 = α. Therefore we have x1 = −x2 = −α, andx3 = 0 . Thus, we get
x =
−αα0
= α
1−10
; α = real number
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
A particular eigenvector is
x(1) =
1−10
2) For λ2 = 1
(A− λ2I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
x1x2x3
= hspace−2mm
1 −3 −10 −2 0−1 1 1
x1x2x3
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
A particular eigenvector is
x(1) =
1−10
2) For λ2 = 1
(A− λ2I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
x1x2x3
= hspace−2mm
1 −3 −10 −2 0−1 1 1
x1x2x3
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
A particular eigenvector is
x(1) =
1−10
2) For λ2 = 1
(A− λ2I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
x1x2x3
= hspace−2mm
1 −3 −10 −2 0−1 1 1
x1x2x3
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
A particular eigenvector is
x(1) =
1−10
2) For λ2 = 1
(A− λ2I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
x1x2x3
= hspace−2mm
1 −3 −10 −2 0−1 1 1
x1x2x3
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
A particular eigenvector is
x(1) =
1−10
2) For λ2 = 1
(A− λ2I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
x1x2x3
= hspace−2mm
1 −3 −10 −2 0−1 1 1
x1x2x3
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
A particular eigenvector is
x(1) =
1−10
2) For λ2 = 1
(A− λ2I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
x1x2x3
=
hspace−2mm
1 −3 −10 −2 0−1 1 1
x1x2x3
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
A particular eigenvector is
x(1) =
1−10
2) For λ2 = 1
(A− λ2I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
x1x2x3
= hspace−2mm
1 −3 −10 −2 0−1 1 1
x1x2x3
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −3 −10 −2 00 −2 0
=
1 −3 −10 −2 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
x1 − 3x2 − x3 = 0 − 2x2 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x3 = α. Therefore we have x1 = x3 = α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −3 −10 −2 00 −2 0
=
1 −3 −10 −2 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
x1 − 3x2 − x3 = 0 − 2x2 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x3 = α. Therefore we have x1 = x3 = α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −3 −10 −2 00 −2 0
=
1 −3 −10 −2 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
x1 − 3x2 − x3 = 0 − 2x2 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x3 = α. Therefore we have x1 = x3 = α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −3 −10 −2 00 −2 0
=
1 −3 −10 −2 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
x1 − 3x2 − x3 = 0 − 2x2 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x3 = α. Therefore we have x1 = x3 = α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −3 −10 −2 00 −2 0
=
1 −3 −10 −2 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
x1 − 3x2 − x3 = 0 − 2x2 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x3 = α. Therefore we have x1 = x3 = α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −3 −10 −2 00 −2 0
=
1 −3 −10 −2 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
x1 − 3x2 − x3 = 0 − 2x2 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x3 = α. Therefore we have x1 = x3 = α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −3 −10 −2 00 −2 0
=
1 −3 −10 −2 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
x1 − 3x2 − x3 = 0 − 2x2 = 0
One equation and two unknowns. Hence,
one of them is freevariable, let’s say x3 = α. Therefore we have x1 = x3 = α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −3 −10 −2 00 −2 0
=
1 −3 −10 −2 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
x1 − 3x2 − x3 = 0 − 2x2 = 0
One equation and two unknowns. Hence, one of them is freevariable,
let’s say x3 = α. Therefore we have x1 = x3 = α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −3 −10 −2 00 −2 0
=
1 −3 −10 −2 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
x1 − 3x2 − x3 = 0 − 2x2 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x3 = α.
Therefore we have x1 = x3 = α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −3 −10 −2 00 −2 0
=
1 −3 −10 −2 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
x1 − 3x2 − x3 = 0 − 2x2 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x3 = α. Therefore we have
x1 = x3 = α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −3 −10 −2 00 −2 0
=
1 −3 −10 −2 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
x1 − 3x2 − x3 = 0 − 2x2 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x3 = α. Therefore we have x1 = x3 = α, and
x2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −3 −10 −2 00 −2 0
=
1 −3 −10 −2 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
x1 − 3x2 − x3 = 0 − 2x2 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x3 = α. Therefore we have x1 = x3 = α, andx2 = 0 .
Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
1 −3 −10 −2 00 −2 0
=
1 −3 −10 −2 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
x1 − 3x2 − x3 = 0 − 2x2 = 0
One equation and two unknowns. Hence, one of them is freevariable, let’s say x3 = α. Therefore we have x1 = x3 = α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x =
α0α
= α
101
; α = real number
A particular eigenvector is given by
x(1) =
101
3) For λ3 = 3
(A− λ3I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x =
α0α
= α
101
; α = real number
A particular eigenvector is given by
x(1) =
101
3) For λ3 = 3
(A− λ3I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x =
α0α
= α
101
; α = real number
A particular eigenvector is given by
x(1) =
101
3) For λ3 = 3
(A− λ3I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x =
α0α
= α
101
; α = real number
A particular eigenvector is given by
x(1) =
101
3) For λ3 = 3
(A− λ3I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x =
α0α
= α
101
; α = real number
A particular eigenvector is given by
x(1) =
101
3) For λ3 = 3
(A− λ3I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x =
α0α
= α
101
; α = real number
A particular eigenvector is given by
x(1) =
101
3) For λ3 = 3
(A− λ3I) x =
2− λ −3 −10 −1− λ 0−1 1 2− λ
=Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
−1 −3 −10 −4 0−1 1 1
=
−1 −3 −10 −4 00 4 0
=
−1 −3 −10 −4 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
−x1 − 3x2 − x3 = 0 − 4x2 = 0
One equation and two unknowns. Hence, one of them is a freevariable, let’s say x3 = α. Therefore we have x1 = −x3 = −α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors−1 −3 −1
0 −4 0−1 1 1
=
−1 −3 −10 −4 00 4 0
=
−1 −3 −10 −4 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
−x1 − 3x2 − x3 = 0 − 4x2 = 0
One equation and two unknowns. Hence, one of them is a freevariable, let’s say x3 = α. Therefore we have x1 = −x3 = −α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors−1 −3 −1
0 −4 0−1 1 1
=
−1 −3 −10 −4 00 4 0
=
−1 −3 −10 −4 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
−x1 − 3x2 − x3 = 0 − 4x2 = 0
One equation and two unknowns. Hence, one of them is a freevariable, let’s say x3 = α. Therefore we have x1 = −x3 = −α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors−1 −3 −1
0 −4 0−1 1 1
=
−1 −3 −10 −4 00 4 0
=
−1 −3 −10 −4 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
−x1 − 3x2 − x3 = 0 − 4x2 = 0
One equation and two unknowns. Hence, one of them is a freevariable, let’s say x3 = α. Therefore we have x1 = −x3 = −α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors−1 −3 −1
0 −4 0−1 1 1
=
−1 −3 −10 −4 00 4 0
=
−1 −3 −10 −4 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
−x1 − 3x2 − x3 = 0 − 4x2 = 0
One equation and two unknowns. Hence, one of them is a freevariable, let’s say x3 = α. Therefore we have x1 = −x3 = −α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors−1 −3 −1
0 −4 0−1 1 1
=
−1 −3 −10 −4 00 4 0
=
−1 −3 −10 −4 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
−x1 − 3x2 − x3 = 0 − 4x2 = 0
One equation and two unknowns. Hence, one of them is a freevariable, let’s say x3 = α. Therefore we have x1 = −x3 = −α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors−1 −3 −1
0 −4 0−1 1 1
=
−1 −3 −10 −4 00 4 0
=
−1 −3 −10 −4 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
−x1 − 3x2 − x3 = 0 − 4x2 = 0
One equation and two unknowns. Hence, one of them is a freevariable, let’s say x3 = α. Therefore we have x1 = −x3 = −α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors−1 −3 −1
0 −4 0−1 1 1
=
−1 −3 −10 −4 00 4 0
=
−1 −3 −10 −4 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
−x1 − 3x2 − x3 = 0 − 4x2 = 0
One equation and two unknowns. Hence,
one of them is a freevariable, let’s say x3 = α. Therefore we have x1 = −x3 = −α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors−1 −3 −1
0 −4 0−1 1 1
=
−1 −3 −10 −4 00 4 0
=
−1 −3 −10 −4 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
−x1 − 3x2 − x3 = 0 − 4x2 = 0
One equation and two unknowns. Hence, one of them is a freevariable,
let’s say x3 = α. Therefore we have x1 = −x3 = −α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors−1 −3 −1
0 −4 0−1 1 1
=
−1 −3 −10 −4 00 4 0
=
−1 −3 −10 −4 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
−x1 − 3x2 − x3 = 0 − 4x2 = 0
One equation and two unknowns. Hence, one of them is a freevariable, let’s say x3 = α.
Therefore we have x1 = −x3 = −α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors−1 −3 −1
0 −4 0−1 1 1
=
−1 −3 −10 −4 00 4 0
=
−1 −3 −10 −4 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
−x1 − 3x2 − x3 = 0 − 4x2 = 0
One equation and two unknowns. Hence, one of them is a freevariable, let’s say x3 = α. Therefore we have
x1 = −x3 = −α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors−1 −3 −1
0 −4 0−1 1 1
=
−1 −3 −10 −4 00 4 0
=
−1 −3 −10 −4 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
−x1 − 3x2 − x3 = 0 − 4x2 = 0
One equation and two unknowns. Hence, one of them is a freevariable, let’s say x3 = α. Therefore we have x1 = −x3 = −α, and
x2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors−1 −3 −1
0 −4 0−1 1 1
=
−1 −3 −10 −4 00 4 0
=
−1 −3 −10 −4 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
−x1 − 3x2 − x3 = 0 − 4x2 = 0
One equation and two unknowns. Hence, one of them is a freevariable, let’s say x3 = α. Therefore we have x1 = −x3 = −α, andx2 = 0 .
Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors−1 −3 −1
0 −4 0−1 1 1
=
−1 −3 −10 −4 00 4 0
=
−1 −3 −10 −4 00 0 0
x1x2x3
=
000
The above system is reduced immediately to the equations
−x1 − 3x2 − x3 = 0 − 4x2 = 0
One equation and two unknowns. Hence, one of them is a freevariable, let’s say x3 = α. Therefore we have x1 = −x3 = −α, andx2 = 0 . Thus, we get
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x =
−α0α
= α
−101
; α = real number
A particular eigenvector is given by
x(1) =
−101
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x =
−α0α
=
α
−101
; α = real number
A particular eigenvector is given by
x(1) =
−101
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x =
−α0α
= α
−101
; α = real number
A particular eigenvector is given by
x(1) =
−101
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x =
−α0α
= α
−101
; α = real number
A particular eigenvector is given by
x(1) =
−101
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x =
−α0α
= α
−101
; α = real number
A particular eigenvector is given by
x(1) =
−101
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Thus, the three linearly independent eigenvectors, are
x(1) =
110
x(2) =
101
x(3) =
− 101
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Thus, the three linearly independent eigenvectors, are
x(1) =
110
x(2) =
101
x(3) =
− 101
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Thus, the three linearly independent eigenvectors, are
x(1) =
110
x(2) =
101
x(3) =
− 101
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Thus, the three linearly independent eigenvectors, are
x(1) =
110
x(2) =
101
x(3) =
− 101
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Thus, the three linearly independent eigenvectors, are
x(1) =
110
x(2) =
101
x(3) =
− 101
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.12
Find the eigenvalues and eigenvectors of the matrix
A =
4 6 61 3 21 −5 −2
Solution
The eigenvalues λ and eigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.12
Find the eigenvalues and eigenvectors of the matrix
A =
4 6 61 3 21 −5 −2
Solution
The eigenvalues λ and eigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.12
Find the eigenvalues and eigenvectors of the matrix
A =
4 6 61 3 21 −5 −2
Solution
The eigenvalues λ and eigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.12
Find the eigenvalues and eigenvectors of the matrix
A =
4 6 61 3 21 −5 −2
Solution
The eigenvalues λ and eigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.12
Find the eigenvalues and eigenvectors of the matrix
A =
4 6 61 3 21 −5 −2
Solution
The eigenvalues λ and eigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.12
Find the eigenvalues and eigenvectors of the matrix
A =
4 6 61 3 21 −5 −2
Solution
The eigenvalues λ and
eigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.12
Find the eigenvalues and eigenvectors of the matrix
A =
4 6 61 3 21 −5 −2
Solution
The eigenvalues λ and eigenvectors x satisfy the equation
(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.12
Find the eigenvalues and eigenvectors of the matrix
A =
4 6 61 3 21 −5 −2
Solution
The eigenvalues λ and eigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 2−1 −5 −2− λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 20 −2− λ −λ
∣∣∣∣∣∣ =
−
∣∣∣∣∣∣1 3− λ 20 −(4− λ)(3− λ) + 6 6− 2(4− λ)0 −2− λ −λ
∣∣∣∣∣∣ = − λ3 + 5λ2 − 8λ+ 4 =
−(λ− 1)(λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 2−1 −5 −2− λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 20 −2− λ −λ
∣∣∣∣∣∣ =
−
∣∣∣∣∣∣1 3− λ 20 −(4− λ)(3− λ) + 6 6− 2(4− λ)0 −2− λ −λ
∣∣∣∣∣∣ = − λ3 + 5λ2 − 8λ+ 4 =
−(λ− 1)(λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 2−1 −5 −2− λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 20 −2− λ −λ
∣∣∣∣∣∣ =
−
∣∣∣∣∣∣1 3− λ 20 −(4− λ)(3− λ) + 6 6− 2(4− λ)0 −2− λ −λ
∣∣∣∣∣∣ = − λ3 + 5λ2 − 8λ+ 4 =
−(λ− 1)(λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 2−1 −5 −2− λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 20 −2− λ −λ
∣∣∣∣∣∣ =
−
∣∣∣∣∣∣1 3− λ 20 −(4− λ)(3− λ) + 6 6− 2(4− λ)0 −2− λ −λ
∣∣∣∣∣∣ = − λ3 + 5λ2 − 8λ+ 4 =
−(λ− 1)(λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 2−1 −5 −2− λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 20 −2− λ −λ
∣∣∣∣∣∣ =
−
∣∣∣∣∣∣1 3− λ 20 −(4− λ)(3− λ) + 6 6− 2(4− λ)0 −2− λ −λ
∣∣∣∣∣∣ = − λ3 + 5λ2 − 8λ+ 4 =
−(λ− 1)(λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 2−1 −5 −2− λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 20 −2− λ −λ
∣∣∣∣∣∣ =
−
∣∣∣∣∣∣1 3− λ 20 −(4− λ)(3− λ) + 6 6− 2(4− λ)0 −2− λ −λ
∣∣∣∣∣∣ = − λ3 + 5λ2 − 8λ+ 4 =
−(λ− 1)(λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 2−1 −5 −2− λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 20 −2− λ −λ
∣∣∣∣∣∣ =
−
∣∣∣∣∣∣1 3− λ 20 −(4− λ)(3− λ) + 6 6− 2(4− λ)0 −2− λ −λ
∣∣∣∣∣∣ = − λ3 + 5λ2 − 8λ+ 4 =
−(λ− 1)(λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 2−1 −5 −2− λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 20 −2− λ −λ
∣∣∣∣∣∣ =
−
∣∣∣∣∣∣1 3− λ 20 −(4− λ)(3− λ) + 6 6− 2(4− λ)0 −2− λ −λ
∣∣∣∣∣∣ = − λ3 + 5λ2 − 8λ+ 4 =
−(λ− 1)(λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 2−1 −5 −2− λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 20 −2− λ −λ
∣∣∣∣∣∣ =
−
∣∣∣∣∣∣1 3− λ 20 −(4− λ)(3− λ) + 6 6− 2(4− λ)0 −2− λ −λ
∣∣∣∣∣∣ = − λ3 + 5λ2 − 8λ+ 4 =
−(λ− 1)(λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 2−1 −5 −2− λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 20 −2− λ −λ
∣∣∣∣∣∣ =
−
∣∣∣∣∣∣1 3− λ 20 −(4− λ)(3− λ) + 6 6− 2(4− λ)0 −2− λ −λ
∣∣∣∣∣∣ =
− λ3 + 5λ2 − 8λ+ 4 =
−(λ− 1)(λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 2−1 −5 −2− λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 20 −2− λ −λ
∣∣∣∣∣∣ =
−
∣∣∣∣∣∣1 3− λ 20 −(4− λ)(3− λ) + 6 6− 2(4− λ)0 −2− λ −λ
∣∣∣∣∣∣ = − λ3 + 5λ2 − 8λ+ 4 =
−(λ− 1)(λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 2−1 −5 −2− λ
∣∣∣∣∣∣ =
∣∣∣∣∣∣4− λ 6 6
1 3− λ 20 −2− λ −λ
∣∣∣∣∣∣ =
−
∣∣∣∣∣∣1 3− λ 20 −(4− λ)(3− λ) + 6 6− 2(4− λ)0 −2− λ −λ
∣∣∣∣∣∣ = − λ3 + 5λ2 − 8λ+ 4 =
−(λ− 1)(λ− 2)2 = 0Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 2, and λ3 = 2 .
1) For λ1 = 1
(A− λ1I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
3 6 61 2 2−1 −5 −3
x1x2x3
=
000
We can reduce this to the equivalent system
3 6 61 2 20 3 1
=
1 2 21 2 20 3 1
=
1 2 21 −3 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 2, and λ3 = 2 .
1) For λ1 = 1
(A− λ1I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
3 6 61 2 2−1 −5 −3
x1x2x3
=
000
We can reduce this to the equivalent system
3 6 61 2 20 3 1
=
1 2 21 2 20 3 1
=
1 2 21 −3 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 2, and λ3 = 2 .
1) For λ1 = 1
(A− λ1I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
3 6 61 2 2−1 −5 −3
x1x2x3
=
000
We can reduce this to the equivalent system
3 6 61 2 20 3 1
=
1 2 21 2 20 3 1
=
1 2 21 −3 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 2, and λ3 = 2 .
1) For λ1 = 1
(A− λ1I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
3 6 61 2 2−1 −5 −3
x1x2x3
=
000
We can reduce this to the equivalent system
3 6 61 2 20 3 1
=
1 2 21 2 20 3 1
=
1 2 21 −3 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 2, and λ3 = 2 .
1) For λ1 = 1
(A− λ1I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
3 6 61 2 2−1 −5 −3
x1x2x3
=
000
We can reduce this to the equivalent system
3 6 61 2 20 3 1
=
1 2 21 2 20 3 1
=
1 2 21 −3 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 2, and λ3 = 2 .
1) For λ1 = 1
(A− λ1I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
3 6 61 2 2−1 −5 −3
x1x2x3
=
000
We can reduce this to the equivalent system
3 6 61 2 20 3 1
=
1 2 21 2 20 3 1
=
1 2 21 −3 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 2, and λ3 = 2 .
1) For λ1 = 1
(A− λ1I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
3 6 61 2 2−1 −5 −3
x1x2x3
=
000
We can reduce this to the equivalent system
3 6 61 2 20 3 1
=
1 2 21 2 20 3 1
=
1 2 21 −3 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 2, and λ3 = 2 .
1) For λ1 = 1
(A− λ1I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
3 6 61 2 2−1 −5 −3
x1x2x3
=
000
We can reduce this to the equivalent system
3 6 61 2 20 3 1
=
1 2 21 2 20 3 1
=
1 2 21 −3 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 2, and λ3 = 2 .
1) For λ1 = 1
(A− λ1I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
3 6 61 2 2−1 −5 −3
x1x2x3
=
000
We can reduce this to the equivalent system
3 6 61 2 20 3 1
=
1 2 21 2 20 3 1
=
1 2 21 −3 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 2, and λ3 = 2 .
1) For λ1 = 1
(A− λ1I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
3 6 61 2 2−1 −5 −3
x1x2x3
=
000
We can reduce this to the equivalent system
3 6 61 2 20 3 1
=
1 2 21 2 20 3 1
=
1 2 21 −3 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 2, and λ3 = 2 .
1) For λ1 = 1
(A− λ1I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
3 6 61 2 2−1 −5 −3
x1x2x3
=
000
We can reduce this to the equivalent system
3 6 61 2 20 3 1
=
1 2 21 2 20 3 1
=
1 2 21 −3 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 2, and λ3 = 2 .
1) For λ1 = 1
(A− λ1I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
x1x2x3
=
3 6 61 2 2−1 −5 −3
x1x2x3
=
000
We can reduce this to the equivalent system
3 6 61 2 20 3 1
=
1 2 21 2 20 3 1
=
1 2 21 −3 −10 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
41−3
2) For λ2 = 2
(A− λ2,3I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
=
2 6 61 1 2−1 −5 −4
x1x2x3
=
2 6 61 1 2−1 −5 −4
=
1 1 20 4 20 −4 −2
=
1 1 20 4 20 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
41−3
2) For λ2 = 2
(A− λ2,3I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
=
2 6 61 1 2−1 −5 −4
x1x2x3
=
2 6 61 1 2−1 −5 −4
=
1 1 20 4 20 −4 −2
=
1 1 20 4 20 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
41−3
2) For λ2 = 2
(A− λ2,3I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
=
2 6 61 1 2−1 −5 −4
x1x2x3
=
2 6 61 1 2−1 −5 −4
=
1 1 20 4 20 −4 −2
=
1 1 20 4 20 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
41−3
2) For λ2 = 2
(A− λ2,3I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
=
2 6 61 1 2−1 −5 −4
x1x2x3
=
2 6 61 1 2−1 −5 −4
=
1 1 20 4 20 −4 −2
=
1 1 20 4 20 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
41−3
2) For λ2 = 2
(A− λ2,3I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
=
2 6 61 1 2−1 −5 −4
x1x2x3
=
2 6 61 1 2−1 −5 −4
=
1 1 20 4 20 −4 −2
=
1 1 20 4 20 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
41−3
2) For λ2 = 2
(A− λ2,3I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
=
2 6 61 1 2−1 −5 −4
x1x2x3
=
2 6 61 1 2−1 −5 −4
=
1 1 20 4 20 −4 −2
=
1 1 20 4 20 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
41−3
2) For λ2 = 2
(A− λ2,3I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
=
2 6 61 1 2−1 −5 −4
x1x2x3
=
2 6 61 1 2−1 −5 −4
=
1 1 20 4 20 −4 −2
=
1 1 20 4 20 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
41−3
2) For λ2 = 2
(A− λ2,3I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
=
2 6 61 1 2−1 −5 −4
x1x2x3
=
2 6 61 1 2−1 −5 −4
=
1 1 20 4 20 −4 −2
=
1 1 20 4 20 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
41−3
2) For λ2 = 2
(A− λ2,3I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
=
2 6 61 1 2−1 −5 −4
x1x2x3
=
2 6 61 1 2−1 −5 −4
=
1 1 20 4 20 −4 −2
=
1 1 20 4 20 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
41−3
2) For λ2 = 2
(A− λ2,3I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
=
2 6 61 1 2−1 −5 −4
x1x2x3
=
2 6 61 1 2−1 −5 −4
=
1 1 20 4 20 −4 −2
=
1 1 20 4 20 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
41−3
2) For λ2 = 2
(A− λ2,3I) x =
4− λ 6 61 3− λ 2−1 −5 −2− λ
=
2 6 61 1 2−1 −5 −4
x1x2x3
=
2 6 61 1 2−1 −5 −4
=
1 1 20 4 20 −4 −2
=
1 1 20 4 20 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 2x3 = 0 4x2 + 2x3 = 0
Two equations and three unknowns. Hence, one of them, is a freevariable, let’s say x3 = α, x2 = 1
2α, and x3 = −2x3 − x2 = −3α .Thus we have
x =
− 3α12α
− 3α
= α
− 312
− 3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 2x3 = 0 4x2 + 2x3 = 0
Two equations and three unknowns. Hence, one of them, is a freevariable, let’s say x3 = α, x2 = 1
2α, and x3 = −2x3 − x2 = −3α .Thus we have
x =
− 3α12α
− 3α
= α
− 312
− 3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 2x3 = 0 4x2 + 2x3 = 0
Two equations and three unknowns. Hence, one of them, is a freevariable, let’s say x3 = α, x2 = 1
2α, and x3 = −2x3 − x2 = −3α .Thus we have
x =
− 3α12α
− 3α
= α
− 312
− 3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 2x3 = 0 4x2 + 2x3 = 0
Two equations and three unknowns. Hence,
one of them, is a freevariable, let’s say x3 = α, x2 = 1
2α, and x3 = −2x3 − x2 = −3α .Thus we have
x =
− 3α12α
− 3α
= α
− 312
− 3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 2x3 = 0 4x2 + 2x3 = 0
Two equations and three unknowns. Hence, one of them, is a freevariable,
let’s say x3 = α, x2 = 12α, and x3 = −2x3 − x2 = −3α .
Thus we have
x =
− 3α12α
− 3α
= α
− 312
− 3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 2x3 = 0 4x2 + 2x3 = 0
Two equations and three unknowns. Hence, one of them, is a freevariable, let’s say x3 = α,
x2 = 12α, and x3 = −2x3 − x2 = −3α .
Thus we have
x =
− 3α12α
− 3α
= α
− 312
− 3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 2x3 = 0 4x2 + 2x3 = 0
Two equations and three unknowns. Hence, one of them, is a freevariable, let’s say x3 = α, x2 = 1
2α, and
x3 = −2x3 − x2 = −3α .Thus we have
x =
− 3α12α
− 3α
= α
− 312
− 3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 2x3 = 0 4x2 + 2x3 = 0
Two equations and three unknowns. Hence, one of them, is a freevariable, let’s say x3 = α, x2 = 1
2α, and x3 = −2x3 − x2 = −3α .
Thus we have
x =
− 3α12α
− 3α
= α
− 312
− 3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 2x3 = 0 4x2 + 2x3 = 0
Two equations and three unknowns. Hence, one of them, is a freevariable, let’s say x3 = α, x2 = 1
2α, and x3 = −2x3 − x2 = −3α .Thus we have
x =
− 3α12α
− 3α
= α
− 312
− 3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 + x2 + 2x3 = 0 4x2 + 2x3 = 0
Two equations and three unknowns. Hence, one of them, is a freevariable, let’s say x3 = α, x2 = 1
2α, and x3 = −2x3 − x2 = −3α .Thus we have
x =
− 3α12α
− 3α
= α
− 312
− 3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there is one linearly independent eigenvector associatedto λ2,3 = 2 , namely,
x(2) =
31
− 2
Therefore, there are just two linearly independent eigenvectors
x(1) =
111
x(2) =
41
− 3
that is, the matrix A is defective
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there is one linearly independent eigenvector associatedto λ2,3 = 2 , namely,
x(2) =
31
− 2
Therefore, there are just two linearly independent eigenvectors
x(1) =
111
x(2) =
41
− 3
that is, the matrix A is defective
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there is one linearly independent eigenvector associatedto λ2,3 = 2 , namely,
x(2) =
31
− 2
Therefore, there are just two linearly independent eigenvectors
x(1) =
111
x(2) =
41
− 3
that is, the matrix A is defective
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there is one linearly independent eigenvector associatedto λ2,3 = 2 , namely,
x(2) =
31
− 2
Therefore, there are just two linearly independent eigenvectors
x(1) =
111
x(2) =
41
− 3
that is, the matrix A is defective
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there is one linearly independent eigenvector associatedto λ2,3 = 2 , namely,
x(2) =
31
− 2
Therefore, there are just two linearly independent eigenvectors
x(1) =
111
x(2) =
41
− 3
that is, the matrix A is defective
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there is one linearly independent eigenvector associatedto λ2,3 = 2 , namely,
x(2) =
31
− 2
Therefore, there are just two linearly independent eigenvectors
x(1) =
111
x(2) =
41
− 3
that is, the matrix A is defective
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there is one linearly independent eigenvector associatedto λ2,3 = 2 , namely,
x(2) =
31
− 2
Therefore, there are just two linearly independent eigenvectors
x(1) =
111
x(2) =
41
− 3
that is, the matrix A is defective
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there is one linearly independent eigenvector associatedto λ2,3 = 2 , namely,
x(2) =
31
− 2
Therefore, there are just two linearly independent eigenvectors
x(1) =
111
x(2) =
41
− 3
that is, the matrix A is defective
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.13
Find the eigenvalues and eigenvectors of the matrix
A =
1 0 02 1 −23 2 1
Solution
The eigenvalues λ andeigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.13
Find the eigenvalues and eigenvectors of the matrix
A =
1 0 02 1 −23 2 1
Solution
The eigenvalues λ andeigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.13
Find the eigenvalues and eigenvectors of the matrix
A =
1 0 02 1 −23 2 1
Solution
The eigenvalues λ andeigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.13
Find the eigenvalues and eigenvectors of the matrix
A =
1 0 02 1 −23 2 1
Solution
The eigenvalues λ andeigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.13
Find the eigenvalues and eigenvectors of the matrix
A =
1 0 02 1 −23 2 1
Solution
The eigenvalues λ andeigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.13
Find the eigenvalues and eigenvectors of the matrix
A =
1 0 02 1 −23 2 1
Solution
The eigenvalues λ and
eigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.13
Find the eigenvalues and eigenvectors of the matrix
A =
1 0 02 1 −23 2 1
Solution
The eigenvalues λ andeigenvectors x satisfy the equation
(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Example 7.13
Find the eigenvalues and eigenvectors of the matrix
A =
1 0 02 1 −23 2 1
Solution
The eigenvalues λ andeigenvectors x satisfy the equation(A− λI) x = 0, or
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣1− λ 0 0
2 1− λ −23 2 1− λ
∣∣∣∣∣∣ = (1− λ)
∣∣∣∣1− λ −22 1− λ
∣∣∣∣ =
(1− λ)3 + 4(1− λ) = (1− λ)(λ2 − 2λ+ 5) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣1− λ 0 0
2 1− λ −23 2 1− λ
∣∣∣∣∣∣ = (1− λ)
∣∣∣∣1− λ −22 1− λ
∣∣∣∣ =
(1− λ)3 + 4(1− λ) = (1− λ)(λ2 − 2λ+ 5) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣1− λ 0 0
2 1− λ −23 2 1− λ
∣∣∣∣∣∣ = (1− λ)
∣∣∣∣1− λ −22 1− λ
∣∣∣∣ =
(1− λ)3 + 4(1− λ) = (1− λ)(λ2 − 2λ+ 5) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣1− λ 0 0
2 1− λ −23 2 1− λ
∣∣∣∣∣∣ = (1− λ)
∣∣∣∣1− λ −22 1− λ
∣∣∣∣ =
(1− λ)3 + 4(1− λ) = (1− λ)(λ2 − 2λ+ 5) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣1− λ 0 0
2 1− λ −23 2 1− λ
∣∣∣∣∣∣ = (1− λ)
∣∣∣∣1− λ −22 1− λ
∣∣∣∣ =
(1− λ)3 + 4(1− λ) = (1− λ)(λ2 − 2λ+ 5) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣1− λ 0 0
2 1− λ −23 2 1− λ
∣∣∣∣∣∣ = (1− λ)
∣∣∣∣1− λ −22 1− λ
∣∣∣∣ =
(1− λ)3 + 4(1− λ) = (1− λ)(λ2 − 2λ+ 5) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣1− λ 0 0
2 1− λ −23 2 1− λ
∣∣∣∣∣∣ =
(1− λ)
∣∣∣∣1− λ −22 1− λ
∣∣∣∣ =
(1− λ)3 + 4(1− λ) = (1− λ)(λ2 − 2λ+ 5) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣1− λ 0 0
2 1− λ −23 2 1− λ
∣∣∣∣∣∣ = (1− λ)
∣∣∣∣1− λ −22 1− λ
∣∣∣∣ =
(1− λ)3 + 4(1− λ) = (1− λ)(λ2 − 2λ+ 5) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣1− λ 0 0
2 1− λ −23 2 1− λ
∣∣∣∣∣∣ = (1− λ)
∣∣∣∣1− λ −22 1− λ
∣∣∣∣ =
(1− λ)3 + 4(1− λ) = (1− λ)(λ2 − 2λ+ 5) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣1− λ 0 0
2 1− λ −23 2 1− λ
∣∣∣∣∣∣ = (1− λ)
∣∣∣∣1− λ −22 1− λ
∣∣∣∣ =
(1− λ)3 + 4(1− λ) =
(1− λ)(λ2 − 2λ+ 5) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
(A− λI) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
000
The eigenvalues are the roots of the equation
|A− λI| =
∣∣∣∣∣∣1− λ 0 0
2 1− λ −23 2 1− λ
∣∣∣∣∣∣ = (1− λ)
∣∣∣∣1− λ −22 1− λ
∣∣∣∣ =
(1− λ)3 + 4(1− λ) = (1− λ)(λ2 − 2λ+ 5) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 1 + 2 i , and λ3 = 1− 2 i .
1) For λ1 = 1
(A− λ1I) x =
1− λ 0 02 1− λ −23 2 1− λ
=
0 0 02 0 −23 2 0
x1x2x3
=
000
We can reduce this to the equivalent system
2 0 −20 0 03 2 0
=
1 0 −10 0 03 2 0
=
1 0 −10 2 40 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 1 + 2 i , and λ3 = 1− 2 i .
1) For λ1 = 1
(A− λ1I) x =
1− λ 0 02 1− λ −23 2 1− λ
=
0 0 02 0 −23 2 0
x1x2x3
=
000
We can reduce this to the equivalent system
2 0 −20 0 03 2 0
=
1 0 −10 0 03 2 0
=
1 0 −10 2 40 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 1 + 2 i , and λ3 = 1− 2 i .
1) For λ1 = 1
(A− λ1I) x =
1− λ 0 02 1− λ −23 2 1− λ
=
0 0 02 0 −23 2 0
x1x2x3
=
000
We can reduce this to the equivalent system
2 0 −20 0 03 2 0
=
1 0 −10 0 03 2 0
=
1 0 −10 2 40 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 1 + 2 i , and λ3 = 1− 2 i .
1) For λ1 = 1
(A− λ1I) x =
1− λ 0 02 1− λ −23 2 1− λ
=
0 0 02 0 −23 2 0
x1x2x3
=
000
We can reduce this to the equivalent system
2 0 −20 0 03 2 0
=
1 0 −10 0 03 2 0
=
1 0 −10 2 40 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 1 + 2 i , and λ3 = 1− 2 i .
1) For λ1 = 1
(A− λ1I) x =
1− λ 0 02 1− λ −23 2 1− λ
=
0 0 02 0 −23 2 0
x1x2x3
=
000
We can reduce this to the equivalent system
2 0 −20 0 03 2 0
=
1 0 −10 0 03 2 0
=
1 0 −10 2 40 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 1 + 2 i , and λ3 = 1− 2 i .
1) For λ1 = 1
(A− λ1I) x =
1− λ 0 02 1− λ −23 2 1− λ
=
0 0 02 0 −23 2 0
x1x2x3
=
000
We can reduce this to the equivalent system
2 0 −20 0 03 2 0
=
1 0 −10 0 03 2 0
=
1 0 −10 2 40 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 1 + 2 i , and λ3 = 1− 2 i .
1) For λ1 = 1
(A− λ1I) x =
1− λ 0 02 1− λ −23 2 1− λ
=
0 0 02 0 −23 2 0
x1x2x3
=
000
We can reduce this to the equivalent system
2 0 −20 0 03 2 0
=
1 0 −10 0 03 2 0
=
1 0 −10 2 40 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 1 + 2 i , and λ3 = 1− 2 i .
1) For λ1 = 1
(A− λ1I) x =
1− λ 0 02 1− λ −23 2 1− λ
=
0 0 02 0 −23 2 0
x1x2x3
=
000
We can reduce this to the equivalent system
2 0 −20 0 03 2 0
=
1 0 −10 0 03 2 0
=
1 0 −10 2 40 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 1 + 2 i , and λ3 = 1− 2 i .
1) For λ1 = 1
(A− λ1I) x =
1− λ 0 02 1− λ −23 2 1− λ
=
0 0 02 0 −23 2 0
x1x2x3
=
000
We can reduce this to the equivalent system
2 0 −20 0 03 2 0
=
1 0 −10 0 03 2 0
=
1 0 −10 2 40 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 1 + 2 i , and λ3 = 1− 2 i .
1) For λ1 = 1
(A− λ1I) x =
1− λ 0 02 1− λ −23 2 1− λ
=
0 0 02 0 −23 2 0
x1x2x3
=
000
We can reduce this to the equivalent system
2 0 −20 0 03 2 0
=
1 0 −10 0 03 2 0
=
1 0 −10 2 40 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 1 + 2 i , and λ3 = 1− 2 i .
1) For λ1 = 1
(A− λ1I) x =
1− λ 0 02 1− λ −23 2 1− λ
=
0 0 02 0 −23 2 0
x1x2x3
=
000
We can reduce this to the equivalent system
2 0 −20 0 03 2 0
=
1 0 −10 0 03 2 0
=
1 0 −10 2 40 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 1 + 2 i , and λ3 = 1− 2 i .
1) For λ1 = 1
(A− λ1I) x =
1− λ 0 02 1− λ −23 2 1− λ
=
0 0 02 0 −23 2 0
x1x2x3
=
000
We can reduce this to the equivalent system
2 0 −20 0 03 2 0
=
1 0 −10 0 03 2 0
=
1 0 −10 2 40 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 1 + 2 i , and λ3 = 1− 2 i .
1) For λ1 = 1
(A− λ1I) x =
1− λ 0 02 1− λ −23 2 1− λ
=
0 0 02 0 −23 2 0
x1x2x3
=
000
We can reduce this to the equivalent system
2 0 −20 0 03 2 0
=
1 0 −10 0 03 2 0
=
1 0 −10 2 40 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 1 + 2 i , and λ3 = 1− 2 i .
1) For λ1 = 1
(A− λ1I) x =
1− λ 0 02 1− λ −23 2 1− λ
=
0 0 02 0 −23 2 0
x1x2x3
=
000
We can reduce this to the equivalent system
2 0 −20 0 03 2 0
=
1 0 −10 0 03 2 0
=
1 0 −10 2 40 0 0
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
1− 3/2
1
2) For λ2 = 1 + 2 i
(A− λ2I) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
−2 i 0 02 −2 i −23 2 −2 i
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
1− 3/2
1
2) For λ2 = 1 + 2 i
(A− λ2I) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
−2 i 0 02 −2 i −23 2 −2 i
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
1− 3/2
1
2) For λ2 = 1 + 2 i
(A− λ2I) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
−2 i 0 02 −2 i −23 2 −2 i
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
1− 3/2
1
2) For λ2 = 1 + 2 i
(A− λ2I) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
−2 i 0 02 −2 i −23 2 −2 i
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
1− 3/2
1
2) For λ2 = 1 + 2 i
(A− λ2I) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
−2 i 0 02 −2 i −23 2 −2 i
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
1− 3/2
1
2) For λ2 = 1 + 2 i
(A− λ2I) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
−2 i 0 02 −2 i −23 2 −2 i
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
1− 3/2
1
2) For λ2 = 1 + 2 i
(A− λ2I) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
−2 i 0 02 −2 i −23 2 −2 i
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
1− 3/2
1
2) For λ2 = 1 + 2 i
(A− λ2I) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
−2 i 0 02 −2 i −23 2 −2 i
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Solving this system yields the eigenvector
x(1) =
1− 3/2
1
2) For λ2 = 1 + 2 i
(A− λ2I) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
−2 i 0 02 −2 i −23 2 −2 i
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0 − 2 ix2 − 2x3 = 0, 2x2 − 2 ix3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is a free veriable, let’s say x2 = α, x3 = −i α . Hence
x =
0α−i α
= α
01
− i
= α
010
− i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0 − 2 ix2 − 2x3 = 0, 2x2 − 2 ix3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is a free veriable, let’s say x2 = α, x3 = −i α . Hence
x =
0α−i α
= α
01
− i
= α
010
− i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0 − 2 ix2 − 2x3 = 0, 2x2 − 2 ix3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is a free veriable, let’s say x2 = α, x3 = −i α . Hence
x =
0α−i α
= α
01
− i
= α
010
− i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0 − 2 ix2 − 2x3 = 0, 2x2 − 2 ix3 = 0
Thus, we have
one equation and two unknowns. Hence, one ofthem 1 is a free veriable, let’s say x2 = α, x3 = −i α . Hence
x =
0α−i α
= α
01
− i
= α
010
− i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0 − 2 ix2 − 2x3 = 0, 2x2 − 2 ix3 = 0
Thus, we have one equation and two unknowns. Hence,
one ofthem 1 is a free veriable, let’s say x2 = α, x3 = −i α . Hence
x =
0α−i α
= α
01
− i
= α
010
− i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0 − 2 ix2 − 2x3 = 0, 2x2 − 2 ix3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is a free veriable,
let’s say x2 = α, x3 = −i α . Hence
x =
0α−i α
= α
01
− i
= α
010
− i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0 − 2 ix2 − 2x3 = 0, 2x2 − 2 ix3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is a free veriable, let’s say x2 = α,
x3 = −i α . Hence
x =
0α−i α
= α
01
− i
= α
010
− i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0 − 2 ix2 − 2x3 = 0, 2x2 − 2 ix3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is a free veriable, let’s say x2 = α, x3 = −i α .
Hence
x =
0α−i α
= α
01
− i
= α
010
− i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0 − 2 ix2 − 2x3 = 0, 2x2 − 2 ix3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is a free veriable, let’s say x2 = α, x3 = −i α . Hence
x =
0α−i α
= α
01
− i
= α
010
− i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0 − 2 ix2 − 2x3 = 0, 2x2 − 2 ix3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is a free veriable, let’s say x2 = α, x3 = −i α . Hence
x =
0α−i α
=
α
01
− i
= α
010
− i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0 − 2 ix2 − 2x3 = 0, 2x2 − 2 ix3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is a free veriable, let’s say x2 = α, x3 = −i α . Hence
x =
0α−i α
= α
01
− i
=
α
010
− i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0 − 2 ix2 − 2x3 = 0, 2x2 − 2 ix3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is a free veriable, let’s say x2 = α, x3 = −i α . Hence
x =
0α−i α
= α
01
− i
= α
010
−
i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0 − 2 ix2 − 2x3 = 0, 2x2 − 2 ix3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is a free veriable, let’s say x2 = α, x3 = −i α . Hence
x =
0α−i α
= α
01
− i
= α
010
− i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there are two real linearly independent eigenvectorassociated to λ2 = 1 + 2 i , namely,
x(2) =
010
x(3) =
001
3) For λ3 = 1− 2 i
(A− λ2I) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
2 i 0 02 2 i −23 2 2 i
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there are two real linearly independent eigenvectorassociated to λ2 = 1 + 2 i , namely,
x(2) =
010
x(3) =
001
3) For λ3 = 1− 2 i
(A− λ2I) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
2 i 0 02 2 i −23 2 2 i
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there are two real linearly independent eigenvectorassociated to λ2 = 1 + 2 i , namely,
x(2) =
010
x(3) =
001
3) For λ3 = 1− 2 i
(A− λ2I) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
2 i 0 02 2 i −23 2 2 i
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there are two real linearly independent eigenvectorassociated to λ2 = 1 + 2 i , namely,
x(2) =
010
x(3) =
001
3) For λ3 = 1− 2 i
(A− λ2I) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
2 i 0 02 2 i −23 2 2 i
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there are two real linearly independent eigenvectorassociated to λ2 = 1 + 2 i , namely,
x(2) =
010
x(3) =
001
3) For λ3 = 1− 2 i
(A− λ2I) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
2 i 0 02 2 i −23 2 2 i
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there are two real linearly independent eigenvectorassociated to λ2 = 1 + 2 i , namely,
x(2) =
010
x(3) =
001
3) For λ3 = 1− 2 i
(A− λ2I) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
2 i 0 02 2 i −23 2 2 i
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there are two real linearly independent eigenvectorassociated to λ2 = 1 + 2 i , namely,
x(2) =
010
x(3) =
001
3) For λ3 = 1− 2 i
(A− λ2I) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
2 i 0 02 2 i −23 2 2 i
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there are two real linearly independent eigenvectorassociated to λ2 = 1 + 2 i , namely,
x(2) =
010
x(3) =
001
3) For λ3 = 1− 2 i
(A− λ2I) x =
1− λ 0 02 1− λ −23 2 1− λ
x1x2x3
=
2 i 0 02 2 i −23 2 2 i
x1x2x3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0; 2i x2 − 2x3 = 0; 2x2 + 2i x3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is free variable, let’s say x2 = α, x3 = i α . Thus we have
x =
0αi α
= α
01i
= α
010
+ i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0; 2i x2 − 2x3 = 0; 2x2 + 2i x3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is free variable, let’s say x2 = α, x3 = i α . Thus we have
x =
0αi α
= α
01i
= α
010
+ i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0; 2i x2 − 2x3 = 0; 2x2 + 2i x3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is free variable, let’s say x2 = α, x3 = i α . Thus we have
x =
0αi α
= α
01i
= α
010
+ i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0; 2i x2 − 2x3 = 0; 2x2 + 2i x3 = 0
Thus, we have one equation and two unknowns. Hence,
one ofthem 1 is free variable, let’s say x2 = α, x3 = i α . Thus we have
x =
0αi α
= α
01i
= α
010
+ i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0; 2i x2 − 2x3 = 0; 2x2 + 2i x3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is free variable,
let’s say x2 = α, x3 = i α . Thus we have
x =
0αi α
= α
01i
= α
010
+ i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0; 2i x2 − 2x3 = 0; 2x2 + 2i x3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is free variable, let’s say x2 = α,
x3 = i α . Thus we have
x =
0αi α
= α
01i
= α
010
+ i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0; 2i x2 − 2x3 = 0; 2x2 + 2i x3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is free variable, let’s say x2 = α, x3 = i α .
Thus we have
x =
0αi α
= α
01i
= α
010
+ i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0; 2i x2 − 2x3 = 0; 2x2 + 2i x3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is free variable, let’s say x2 = α, x3 = i α . Thus we have
x =
0αi α
= α
01i
= α
010
+ i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0; 2i x2 − 2x3 = 0; 2x2 + 2i x3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is free variable, let’s say x2 = α, x3 = i α . Thus we have
x =
0αi α
=
α
01i
= α
010
+ i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0; 2i x2 − 2x3 = 0; 2x2 + 2i x3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is free variable, let’s say x2 = α, x3 = i α . Thus we have
x =
0αi α
= α
01i
=
α
010
+ i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0; 2i x2 − 2x3 = 0; 2x2 + 2i x3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is free variable, let’s say x2 = α, x3 = i α . Thus we have
x =
0αi α
= α
01i
= α
010
+
i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0; 2i x2 − 2x3 = 0; 2x2 + 2i x3 = 0
Thus, we have one equation and two unknowns. Hence, one ofthem 1 is free variable, let’s say x2 = α, x3 = i α . Thus we have
x =
0αi α
= α
01i
= α
010
+ i
001
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there is two real linearly independent eigenvectorassociated to λ2 = 1 + 2 i , namely,
x(2) =
010
x(3) =
001
Hence, we have three linearly independent eigenvectors, namely
x(1) =
1− 3/2
1
x(2) =
010
x(3) =
001
that is, the matrix A is Non-defective
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there is two real linearly independent eigenvectorassociated to λ2 = 1 + 2 i , namely,
x(2) =
010
x(3) =
001
Hence, we have three linearly independent eigenvectors, namely
x(1) =
1− 3/2
1
x(2) =
010
x(3) =
001
that is, the matrix A is Non-defective
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there is two real linearly independent eigenvectorassociated to λ2 = 1 + 2 i , namely,
x(2) =
010
x(3) =
001
Hence, we have three linearly independent eigenvectors, namely
x(1) =
1− 3/2
1
x(2) =
010
x(3) =
001
that is, the matrix A is Non-defective
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there is two real linearly independent eigenvectorassociated to λ2 = 1 + 2 i , namely,
x(2) =
010
x(3) =
001
Hence, we have three linearly independent eigenvectors, namely
x(1) =
1− 3/2
1
x(2) =
010
x(3) =
001
that is, the matrix A is Non-defective
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there is two real linearly independent eigenvectorassociated to λ2 = 1 + 2 i , namely,
x(2) =
010
x(3) =
001
Hence,
we have three linearly independent eigenvectors, namely
x(1) =
1− 3/2
1
x(2) =
010
x(3) =
001
that is, the matrix A is Non-defective
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there is two real linearly independent eigenvectorassociated to λ2 = 1 + 2 i , namely,
x(2) =
010
x(3) =
001
Hence, we have three linearly independent eigenvectors, namely
x(1) =
1− 3/2
1
x(2) =
010
x(3) =
001
that is, the matrix A is Non-defective
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there is two real linearly independent eigenvectorassociated to λ2 = 1 + 2 i , namely,
x(2) =
010
x(3) =
001
Hence, we have three linearly independent eigenvectors, namely
x(1) =
1− 3/2
1
x(2) =
010
x(3) =
001
that is, the matrix A is Non-defective
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there is two real linearly independent eigenvectorassociated to λ2 = 1 + 2 i , namely,
x(2) =
010
x(3) =
001
Hence, we have three linearly independent eigenvectors, namely
x(1) =
1− 3/2
1
x(2) =
010
x(3) =
001
that is, the matrix A is Non-defective
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there is two real linearly independent eigenvectorassociated to λ2 = 1 + 2 i , namely,
x(2) =
010
x(3) =
001
Hence, we have three linearly independent eigenvectors, namely
x(1) =
1− 3/2
1
x(2) =
010
x(3) =
001
that is, the matrix A is Non-defective
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
In this way, there is two real linearly independent eigenvectorassociated to λ2 = 1 + 2 i , namely,
x(2) =
010
x(3) =
001
Hence, we have three linearly independent eigenvectors, namely
x(1) =
1− 3/2
1
x(2) =
010
x(3) =
001
that is, the matrix A is Non-defective
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
OBS
Let A be a real-valued n × n matrix. If x(1) = x(R) ± i x(I ) arecomplex eigenvectors of the matrix A with complex eigenvaluesλ = u ± i v . then, x(R) and x(I ) are two real eigenvectors for Awith eigenvalues λ = u ± i v
Finally, let’s introduce another concept from linear algebra
The Dot Product
For y = (y1, y2, ..., yn), x = (x1, x2, ..., xn) ∈ R, define the dotproduct or inner product or scalar product.as
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
OBS
Let A be a real-valued n × n matrix. If x(1) = x(R) ± i x(I ) arecomplex eigenvectors of the matrix A with complex eigenvaluesλ = u ± i v . then, x(R) and x(I ) are two real eigenvectors for Awith eigenvalues λ = u ± i v
Finally, let’s introduce another concept from linear algebra
The Dot Product
For y = (y1, y2, ..., yn), x = (x1, x2, ..., xn) ∈ R, define the dotproduct or inner product or scalar product.as
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
OBS
Let A be a real-valued n × n matrix.
If x(1) = x(R) ± i x(I ) arecomplex eigenvectors of the matrix A with complex eigenvaluesλ = u ± i v . then, x(R) and x(I ) are two real eigenvectors for Awith eigenvalues λ = u ± i v
Finally, let’s introduce another concept from linear algebra
The Dot Product
For y = (y1, y2, ..., yn), x = (x1, x2, ..., xn) ∈ R, define the dotproduct or inner product or scalar product.as
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
OBS
Let A be a real-valued n × n matrix. If x(1) = x(R) ± i x(I )
arecomplex eigenvectors of the matrix A with complex eigenvaluesλ = u ± i v . then, x(R) and x(I ) are two real eigenvectors for Awith eigenvalues λ = u ± i v
Finally, let’s introduce another concept from linear algebra
The Dot Product
For y = (y1, y2, ..., yn), x = (x1, x2, ..., xn) ∈ R, define the dotproduct or inner product or scalar product.as
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
OBS
Let A be a real-valued n × n matrix. If x(1) = x(R) ± i x(I ) arecomplex eigenvectors of the matrix A
with complex eigenvaluesλ = u ± i v . then, x(R) and x(I ) are two real eigenvectors for Awith eigenvalues λ = u ± i v
Finally, let’s introduce another concept from linear algebra
The Dot Product
For y = (y1, y2, ..., yn), x = (x1, x2, ..., xn) ∈ R, define the dotproduct or inner product or scalar product.as
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
OBS
Let A be a real-valued n × n matrix. If x(1) = x(R) ± i x(I ) arecomplex eigenvectors of the matrix A with complex eigenvaluesλ = u ± i v . then,
x(R) and x(I ) are two real eigenvectors for Awith eigenvalues λ = u ± i v
Finally, let’s introduce another concept from linear algebra
The Dot Product
For y = (y1, y2, ..., yn), x = (x1, x2, ..., xn) ∈ R, define the dotproduct or inner product or scalar product.as
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
OBS
Let A be a real-valued n × n matrix. If x(1) = x(R) ± i x(I ) arecomplex eigenvectors of the matrix A with complex eigenvaluesλ = u ± i v . then, x(R) and x(I )
are two real eigenvectors for Awith eigenvalues λ = u ± i v
Finally, let’s introduce another concept from linear algebra
The Dot Product
For y = (y1, y2, ..., yn), x = (x1, x2, ..., xn) ∈ R, define the dotproduct or inner product or scalar product.as
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
OBS
Let A be a real-valued n × n matrix. If x(1) = x(R) ± i x(I ) arecomplex eigenvectors of the matrix A with complex eigenvaluesλ = u ± i v . then, x(R) and x(I ) are two real eigenvectors for A
with eigenvalues λ = u ± i v
Finally, let’s introduce another concept from linear algebra
The Dot Product
For y = (y1, y2, ..., yn), x = (x1, x2, ..., xn) ∈ R, define the dotproduct or inner product or scalar product.as
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
OBS
Let A be a real-valued n × n matrix. If x(1) = x(R) ± i x(I ) arecomplex eigenvectors of the matrix A with complex eigenvaluesλ = u ± i v . then, x(R) and x(I ) are two real eigenvectors for Awith eigenvalues λ = u ± i v
Finally, let’s introduce another concept from linear algebra
The Dot Product
For y = (y1, y2, ..., yn), x = (x1, x2, ..., xn) ∈ R, define the dotproduct or inner product or scalar product.as
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
OBS
Let A be a real-valued n × n matrix. If x(1) = x(R) ± i x(I ) arecomplex eigenvectors of the matrix A with complex eigenvaluesλ = u ± i v . then, x(R) and x(I ) are two real eigenvectors for Awith eigenvalues λ = u ± i v
Finally, let’s introduce another concept from linear algebra
The Dot Product
For y = (y1, y2, ..., yn), x = (x1, x2, ..., xn) ∈ R, define the dotproduct or inner product or scalar product.as
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
OBS
Let A be a real-valued n × n matrix. If x(1) = x(R) ± i x(I ) arecomplex eigenvectors of the matrix A with complex eigenvaluesλ = u ± i v . then, x(R) and x(I ) are two real eigenvectors for Awith eigenvalues λ = u ± i v
Finally, let’s introduce another concept from linear algebra
The Dot Product
For y = (y1, y2, ..., yn), x = (x1, x2, ..., xn) ∈ R, define the dotproduct or inner product or scalar product.as
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
OBS
Let A be a real-valued n × n matrix. If x(1) = x(R) ± i x(I ) arecomplex eigenvectors of the matrix A with complex eigenvaluesλ = u ± i v . then, x(R) and x(I ) are two real eigenvectors for Awith eigenvalues λ = u ± i v
Finally, let’s introduce another concept from linear algebra
The Dot Product
For y = (y1, y2, ..., yn), x = (x1, x2, ..., xn) ∈ R, define the dotproduct or inner product or scalar product.as
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x · y = < x, y >=(x1 x2 . . . xn
)y1y2...yn
= x1y1 + x2y2 + ...+ xnyn
OBS
a) x and y are said to be orthogonal if < x, y >= 0 .
b) Orthogonal nonzero vectors are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x · y =
< x, y >=(x1 x2 . . . xn
)y1y2...yn
= x1y1 + x2y2 + ...+ xnyn
OBS
a) x and y are said to be orthogonal if < x, y >= 0 .
b) Orthogonal nonzero vectors are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x · y = < x, y >=
(x1 x2 . . . xn
)y1y2...yn
= x1y1 + x2y2 + ...+ xnyn
OBS
a) x and y are said to be orthogonal if < x, y >= 0 .
b) Orthogonal nonzero vectors are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x · y = < x, y >=(x1 x2 . . . xn
)
y1y2...yn
= x1y1 + x2y2 + ...+ xnyn
OBS
a) x and y are said to be orthogonal if < x, y >= 0 .
b) Orthogonal nonzero vectors are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x · y = < x, y >=(x1 x2 . . . xn
)y1y2...yn
=
x1y1 + x2y2 + ...+ xnyn
OBS
a) x and y are said to be orthogonal if < x, y >= 0 .
b) Orthogonal nonzero vectors are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x · y = < x, y >=(x1 x2 . . . xn
)y1y2...yn
= x1y1 + x2y2 + ...+ xnyn
OBS
a) x and y are said to be orthogonal if < x, y >= 0 .
b) Orthogonal nonzero vectors are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x · y = < x, y >=(x1 x2 . . . xn
)y1y2...yn
= x1y1 + x2y2 + ...+ xnyn
OBS
a) x and y are said to be orthogonal if < x, y >= 0 .
b) Orthogonal nonzero vectors are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x · y = < x, y >=(x1 x2 . . . xn
)y1y2...yn
= x1y1 + x2y2 + ...+ xnyn
OBS
a) x and y are said to be orthogonal if < x, y >= 0 .
b) Orthogonal nonzero vectors are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
x · y = < x, y >=(x1 x2 . . . xn
)y1y2...yn
= x1y1 + x2y2 + ...+ xnyn
OBS
a) x and y are said to be orthogonal if < x, y >= 0 .
b) Orthogonal nonzero vectors are linearly independent.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.10
Let A be an n × n matrix. If A is symetric, ( A = AT ) then
1) All eigenvalues are real.
2) A is always Nondefective.
3) The eigenvectors corresponding to different eigenvalues areorthogonal, thus if λ1, λ2, ..., λn are all simple, v1, v2, ..., vn forman orthogonal set.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.10
Let A be an n × n matrix. If A is symetric, ( A = AT ) then
1) All eigenvalues are real.
2) A is always Nondefective.
3) The eigenvectors corresponding to different eigenvalues areorthogonal, thus if λ1, λ2, ..., λn are all simple, v1, v2, ..., vn forman orthogonal set.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.10
Let A be an n × n matrix. If A is symetric, ( A = AT ) then
1) All eigenvalues are real.
2) A is always Nondefective.
3) The eigenvectors corresponding to different eigenvalues areorthogonal, thus if λ1, λ2, ..., λn are all simple, v1, v2, ..., vn forman orthogonal set.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.10
Let A be an n × n matrix. If A is symetric, ( A = AT ) then
1) All eigenvalues are real.
2) A is always Nondefective.
3) The eigenvectors corresponding to different eigenvalues areorthogonal, thus if λ1, λ2, ..., λn are all simple, v1, v2, ..., vn forman orthogonal set.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.10
Let A be an n × n matrix. If A is symetric, ( A = AT ) then
1) All eigenvalues are real.
2) A is always Nondefective.
3) The eigenvectors corresponding to different eigenvalues areorthogonal, thus if λ1, λ2, ..., λn are all simple, v1, v2, ..., vn forman orthogonal set.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.10
Let A be an n × n matrix. If A is symetric, ( A = AT ) then
1) All eigenvalues are real.
2) A is always Nondefective.
3) The eigenvectors corresponding to different eigenvalues areorthogonal, thus if λ1, λ2, ..., λn are all simple, v1, v2, ..., vn forman orthogonal set.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.10
Let A be an n × n matrix. If A is symetric, ( A = AT ) then
1) All eigenvalues are real.
2) A is always Nondefective.
3) The eigenvectors corresponding to different eigenvalues areorthogonal, thus
if λ1, λ2, ..., λn are all simple, v1, v2, ..., vn forman orthogonal set.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.10
Let A be an n × n matrix. If A is symetric, ( A = AT ) then
1) All eigenvalues are real.
2) A is always Nondefective.
3) The eigenvectors corresponding to different eigenvalues areorthogonal, thus if λ1, λ2, ..., λn
are all simple, v1, v2, ..., vn forman orthogonal set.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.10
Let A be an n × n matrix. If A is symetric, ( A = AT ) then
1) All eigenvalues are real.
2) A is always Nondefective.
3) The eigenvectors corresponding to different eigenvalues areorthogonal, thus if λ1, λ2, ..., λn are all simple,
v1, v2, ..., vn forman orthogonal set.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; LinearIndependence, Eigenvalues, Eigenvectors
Theorem 7.10
Let A be an n × n matrix. If A is symetric, ( A = AT ) then
1) All eigenvalues are real.
2) A is always Nondefective.
3) The eigenvectors corresponding to different eigenvalues areorthogonal, thus if λ1, λ2, ..., λn are all simple, v1, v2, ..., vn forman orthogonal set.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
The general theory of a system of n first order linear equations
x ′1 = p11x1 + p12x2 + . . .+ p1nxn + g1(t)x ′2 = p21x1 + p22x2 + . . .+ p2nxn + g2(t)...
...x ′n = pn1x1 + pn2x2 + . . .+ pnnxn + gn(t)
or
X′ = P(t)X + g(t)
closely parallels that of a single linear equation of nth order.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
The general theory of a system of n first order linear equations
x ′1 = p11x1 + p12x2 + . . .+ p1nxn + g1(t)x ′2 = p21x1 + p22x2 + . . .+ p2nxn + g2(t)...
...x ′n = pn1x1 + pn2x2 + . . .+ pnnxn + gn(t)
or
X′ = P(t)X + g(t)
closely parallels that of a single linear equation of nth order.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
The general theory of a system of n first order linear equations
x ′1 = p11x1 + p12x2 + . . .+ p1nxn + g1(t)x ′2 = p21x1 + p22x2 + . . .+ p2nxn + g2(t)...
...x ′n = pn1x1 + pn2x2 + . . .+ pnnxn + gn(t)
or
X′ = P(t)X + g(t)
closely parallels that of a single linear equation of nth order.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
The general theory of a system of n first order linear equations
x ′1 = p11x1 + p12x2 + . . .+ p1nxn + g1(t)x ′2 = p21x1 + p22x2 + . . .+ p2nxn + g2(t)...
...x ′n = pn1x1 + pn2x2 + . . .+ pnnxn + gn(t)
or
X′ = P(t)X + g(t)
closely parallels that of a single linear equation of nth order.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
The general theory of a system of n first order linear equations
x ′1 = p11x1 + p12x2 + . . .+ p1nxn + g1(t)x ′2 = p21x1 + p22x2 + . . .+ p2nxn + g2(t)...
...x ′n = pn1x1 + pn2x2 + . . .+ pnnxn + gn(t)
or
X′ = P(t)X + g(t)
closely parallels that of a single linear equation of nth order.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
The general theory of a system of n first order linear equations
x ′1 = p11x1 + p12x2 + . . .+ p1nxn + g1(t)x ′2 = p21x1 + p22x2 + . . .+ p2nxn + g2(t)...
...x ′n = pn1x1 + pn2x2 + . . .+ pnnxn + gn(t)
or
X′ = P(t)X + g(t)
closely parallels that of a single linear equation of nth order.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
The general theory of a system of n first order linear equations
x ′1 = p11x1 + p12x2 + . . .+ p1nxn + g1(t)x ′2 = p21x1 + p22x2 + . . .+ p2nxn + g2(t)...
...x ′n = pn1x1 + pn2x2 + . . .+ pnnxn + gn(t)
or
X′ = P(t)X + g(t)
closely parallels that of a single linear equation of nth order.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
we assume that P and g are continuous on some intervalα < t < β; that is, each of the scalar functionsp11, ..., pnn, g1, ..., gn is continuous there.
Theorem 7.4
If the vector functions x(1) and x(2) are solutions of thehomogeneus system ( g(t) = 0 ) then the linear combinationc1x(1) + c2x(2) is also a solution for any constants c1 and c2.
This is the principle of superposition
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
we assume that P and g are continuous on some intervalα < t < β;
that is, each of the scalar functionsp11, ..., pnn, g1, ..., gn is continuous there.
Theorem 7.4
If the vector functions x(1) and x(2) are solutions of thehomogeneus system ( g(t) = 0 ) then the linear combinationc1x(1) + c2x(2) is also a solution for any constants c1 and c2.
This is the principle of superposition
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
we assume that P and g are continuous on some intervalα < t < β; that is, each of the scalar functionsp11, ..., pnn, g1, ..., gn is continuous there.
Theorem 7.4
If the vector functions x(1) and x(2) are solutions of thehomogeneus system ( g(t) = 0 ) then the linear combinationc1x(1) + c2x(2) is also a solution for any constants c1 and c2.
This is the principle of superposition
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
we assume that P and g are continuous on some intervalα < t < β; that is, each of the scalar functionsp11, ..., pnn, g1, ..., gn is continuous there.
Theorem 7.4
If the vector functions x(1) and x(2) are solutions of thehomogeneus system ( g(t) = 0 ) then the linear combinationc1x(1) + c2x(2) is also a solution for any constants c1 and c2.
This is the principle of superposition
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
we assume that P and g are continuous on some intervalα < t < β; that is, each of the scalar functionsp11, ..., pnn, g1, ..., gn is continuous there.
Theorem 7.4
If the vector functions x(1) and x(2)
are solutions of thehomogeneus system ( g(t) = 0 ) then the linear combinationc1x(1) + c2x(2) is also a solution for any constants c1 and c2.
This is the principle of superposition
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
we assume that P and g are continuous on some intervalα < t < β; that is, each of the scalar functionsp11, ..., pnn, g1, ..., gn is continuous there.
Theorem 7.4
If the vector functions x(1) and x(2) are solutions of thehomogeneus system ( g(t) = 0 )
then the linear combinationc1x(1) + c2x(2) is also a solution for any constants c1 and c2.
This is the principle of superposition
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
we assume that P and g are continuous on some intervalα < t < β; that is, each of the scalar functionsp11, ..., pnn, g1, ..., gn is continuous there.
Theorem 7.4
If the vector functions x(1) and x(2) are solutions of thehomogeneus system ( g(t) = 0 ) then the linear combinationc1x(1) + c2x(2)
is also a solution for any constants c1 and c2.
This is the principle of superposition
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
we assume that P and g are continuous on some intervalα < t < β; that is, each of the scalar functionsp11, ..., pnn, g1, ..., gn is continuous there.
Theorem 7.4
If the vector functions x(1) and x(2) are solutions of thehomogeneus system ( g(t) = 0 ) then the linear combinationc1x(1) + c2x(2) is also a solution
for any constants c1 and c2.
This is the principle of superposition
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
we assume that P and g are continuous on some intervalα < t < β; that is, each of the scalar functionsp11, ..., pnn, g1, ..., gn is continuous there.
Theorem 7.4
If the vector functions x(1) and x(2) are solutions of thehomogeneus system ( g(t) = 0 ) then the linear combinationc1x(1) + c2x(2) is also a solution for any constants c1 and c2.
This is the principle of superposition
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
we assume that P and g are continuous on some intervalα < t < β; that is, each of the scalar functionsp11, ..., pnn, g1, ..., gn is continuous there.
Theorem 7.4
If the vector functions x(1) and x(2) are solutions of thehomogeneus system ( g(t) = 0 ) then the linear combinationc1x(1) + c2x(2) is also a solution for any constants c1 and c2.
This is the principle of superposition
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
By repeated application of Theorem, we can conclude that ifx(1), ..., x(k) are solutions of the homogeneous system, then
c1x(1) + ...+ ckx(k)
is also a solution for any constants c1, ..., ck .
Theorem 7.5
If the vector functions x(1), ..., x(n) are linearly independentsolutions of the homogeneous system for each point in the intervalα < t < β, then each solution x = φ(t) of the homogeneoussystem can be expressed as a linear combination of x(1), ..., x(n) inexactly one way.
φ(t) = c1x(1) + ...+ ckx(k)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
By repeated application of Theorem, we can conclude that ifx(1), ..., x(k) are solutions of the homogeneous system, then
c1x(1) + ...+ ckx(k)
is also a solution for any constants c1, ..., ck .
Theorem 7.5
If the vector functions x(1), ..., x(n) are linearly independentsolutions of the homogeneous system for each point in the intervalα < t < β, then each solution x = φ(t) of the homogeneoussystem can be expressed as a linear combination of x(1), ..., x(n) inexactly one way.
φ(t) = c1x(1) + ...+ ckx(k)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
By repeated application of Theorem, we can conclude that ifx(1), ..., x(k) are solutions of the homogeneous system, then
c1x(1) + ...+ ckx(k)
is also a solution for any constants c1, ..., ck .
Theorem 7.5
If the vector functions x(1), ..., x(n) are linearly independentsolutions of the homogeneous system for each point in the intervalα < t < β, then each solution x = φ(t) of the homogeneoussystem can be expressed as a linear combination of x(1), ..., x(n) inexactly one way.
φ(t) = c1x(1) + ...+ ckx(k)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
By repeated application of Theorem, we can conclude that ifx(1), ..., x(k) are solutions of the homogeneous system, then
c1x(1) + ...+ ckx(k)
is also a solution for any constants c1, ..., ck .
Theorem 7.5
If the vector functions x(1), ..., x(n) are linearly independentsolutions of the homogeneous system for each point in the intervalα < t < β, then each solution x = φ(t) of the homogeneoussystem can be expressed as a linear combination of x(1), ..., x(n) inexactly one way.
φ(t) = c1x(1) + ...+ ckx(k)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
By repeated application of Theorem, we can conclude that ifx(1), ..., x(k) are solutions of the homogeneous system, then
c1x(1) + ...+ ckx(k)
is also a solution for any constants c1, ..., ck .
Theorem 7.5
If the vector functions x(1), ..., x(n) are linearly independentsolutions of the homogeneous system for each point in the intervalα < t < β, then each solution x = φ(t) of the homogeneoussystem can be expressed as a linear combination of x(1), ..., x(n) inexactly one way.
φ(t) = c1x(1) + ...+ ckx(k)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
By repeated application of Theorem, we can conclude that ifx(1), ..., x(k) are solutions of the homogeneous system, then
c1x(1) + ...+ ckx(k)
is also a solution for any constants c1, ..., ck .
Theorem 7.5
If the vector functions x(1), ..., x(n) are
linearly independentsolutions of the homogeneous system for each point in the intervalα < t < β, then each solution x = φ(t) of the homogeneoussystem can be expressed as a linear combination of x(1), ..., x(n) inexactly one way.
φ(t) = c1x(1) + ...+ ckx(k)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
By repeated application of Theorem, we can conclude that ifx(1), ..., x(k) are solutions of the homogeneous system, then
c1x(1) + ...+ ckx(k)
is also a solution for any constants c1, ..., ck .
Theorem 7.5
If the vector functions x(1), ..., x(n) are linearly independentsolutions of the homogeneous system
for each point in the intervalα < t < β, then each solution x = φ(t) of the homogeneoussystem can be expressed as a linear combination of x(1), ..., x(n) inexactly one way.
φ(t) = c1x(1) + ...+ ckx(k)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
By repeated application of Theorem, we can conclude that ifx(1), ..., x(k) are solutions of the homogeneous system, then
c1x(1) + ...+ ckx(k)
is also a solution for any constants c1, ..., ck .
Theorem 7.5
If the vector functions x(1), ..., x(n) are linearly independentsolutions of the homogeneous system for each point in the intervalα < t < β,
then each solution x = φ(t) of the homogeneoussystem can be expressed as a linear combination of x(1), ..., x(n) inexactly one way.
φ(t) = c1x(1) + ...+ ckx(k)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
By repeated application of Theorem, we can conclude that ifx(1), ..., x(k) are solutions of the homogeneous system, then
c1x(1) + ...+ ckx(k)
is also a solution for any constants c1, ..., ck .
Theorem 7.5
If the vector functions x(1), ..., x(n) are linearly independentsolutions of the homogeneous system for each point in the intervalα < t < β, then each solution x = φ(t) of the homogeneoussystem
can be expressed as a linear combination of x(1), ..., x(n) inexactly one way.
φ(t) = c1x(1) + ...+ ckx(k)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
By repeated application of Theorem, we can conclude that ifx(1), ..., x(k) are solutions of the homogeneous system, then
c1x(1) + ...+ ckx(k)
is also a solution for any constants c1, ..., ck .
Theorem 7.5
If the vector functions x(1), ..., x(n) are linearly independentsolutions of the homogeneous system for each point in the intervalα < t < β, then each solution x = φ(t) of the homogeneoussystem can be expressed as a linear combination of
x(1), ..., x(n) inexactly one way.
φ(t) = c1x(1) + ...+ ckx(k)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
By repeated application of Theorem, we can conclude that ifx(1), ..., x(k) are solutions of the homogeneous system, then
c1x(1) + ...+ ckx(k)
is also a solution for any constants c1, ..., ck .
Theorem 7.5
If the vector functions x(1), ..., x(n) are linearly independentsolutions of the homogeneous system for each point in the intervalα < t < β, then each solution x = φ(t) of the homogeneoussystem can be expressed as a linear combination of x(1), ..., x(n) inexactly one way.
φ(t) = c1x(1) + ...+ ckx(k)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
If the constants c1, ..., cn are thought of as arbitrary, then theabove equation includes all solutions of the system, and it iscustomary to call it the general solution.
Any set of solutions x(1), ..., x(n) of the homogeneus system that islinearly independent at each point in the interval α < t < βis saidto be a fundamental set of solutions for that interval.
Theorem 7.6
If x(1), ..., x(n) are solutions of the homogeneus system on theinterval α < t < β, then in this interval W [x(1), ..., x(n)] given by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
If the constants c1, ..., cn are thought of as arbitrary,
then theabove equation includes all solutions of the system, and it iscustomary to call it the general solution.
Any set of solutions x(1), ..., x(n) of the homogeneus system that islinearly independent at each point in the interval α < t < βis saidto be a fundamental set of solutions for that interval.
Theorem 7.6
If x(1), ..., x(n) are solutions of the homogeneus system on theinterval α < t < β, then in this interval W [x(1), ..., x(n)] given by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
If the constants c1, ..., cn are thought of as arbitrary, then theabove equation
includes all solutions of the system, and it iscustomary to call it the general solution.
Any set of solutions x(1), ..., x(n) of the homogeneus system that islinearly independent at each point in the interval α < t < βis saidto be a fundamental set of solutions for that interval.
Theorem 7.6
If x(1), ..., x(n) are solutions of the homogeneus system on theinterval α < t < β, then in this interval W [x(1), ..., x(n)] given by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
If the constants c1, ..., cn are thought of as arbitrary, then theabove equation includes all solutions of the system, and
it iscustomary to call it the general solution.
Any set of solutions x(1), ..., x(n) of the homogeneus system that islinearly independent at each point in the interval α < t < βis saidto be a fundamental set of solutions for that interval.
Theorem 7.6
If x(1), ..., x(n) are solutions of the homogeneus system on theinterval α < t < β, then in this interval W [x(1), ..., x(n)] given by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
If the constants c1, ..., cn are thought of as arbitrary, then theabove equation includes all solutions of the system, and it iscustomary to call it the general solution.
Any set of solutions x(1), ..., x(n) of the homogeneus system that islinearly independent at each point in the interval α < t < βis saidto be a fundamental set of solutions for that interval.
Theorem 7.6
If x(1), ..., x(n) are solutions of the homogeneus system on theinterval α < t < β, then in this interval W [x(1), ..., x(n)] given by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
If the constants c1, ..., cn are thought of as arbitrary, then theabove equation includes all solutions of the system, and it iscustomary to call it the general solution.
Any set of solutions x(1), ..., x(n) of the homogeneus system
that islinearly independent at each point in the interval α < t < βis saidto be a fundamental set of solutions for that interval.
Theorem 7.6
If x(1), ..., x(n) are solutions of the homogeneus system on theinterval α < t < β, then in this interval W [x(1), ..., x(n)] given by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
If the constants c1, ..., cn are thought of as arbitrary, then theabove equation includes all solutions of the system, and it iscustomary to call it the general solution.
Any set of solutions x(1), ..., x(n) of the homogeneus system that islinearly independent at each point in the interval α < t < β
is saidto be a fundamental set of solutions for that interval.
Theorem 7.6
If x(1), ..., x(n) are solutions of the homogeneus system on theinterval α < t < β, then in this interval W [x(1), ..., x(n)] given by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
If the constants c1, ..., cn are thought of as arbitrary, then theabove equation includes all solutions of the system, and it iscustomary to call it the general solution.
Any set of solutions x(1), ..., x(n) of the homogeneus system that islinearly independent at each point in the interval α < t < βis saidto be a fundamental set of solutions for that interval.
Theorem 7.6
If x(1), ..., x(n) are solutions of the homogeneus system on theinterval α < t < β, then in this interval W [x(1), ..., x(n)] given by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
If the constants c1, ..., cn are thought of as arbitrary, then theabove equation includes all solutions of the system, and it iscustomary to call it the general solution.
Any set of solutions x(1), ..., x(n) of the homogeneus system that islinearly independent at each point in the interval α < t < βis saidto be a fundamental set of solutions for that interval.
Theorem 7.6
If x(1), ..., x(n) are solutions of the homogeneus system on theinterval α < t < β, then in this interval W [x(1), ..., x(n)] given by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
If the constants c1, ..., cn are thought of as arbitrary, then theabove equation includes all solutions of the system, and it iscustomary to call it the general solution.
Any set of solutions x(1), ..., x(n) of the homogeneus system that islinearly independent at each point in the interval α < t < βis saidto be a fundamental set of solutions for that interval.
Theorem 7.6
If x(1), ..., x(n)
are solutions of the homogeneus system on theinterval α < t < β, then in this interval W [x(1), ..., x(n)] given by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
If the constants c1, ..., cn are thought of as arbitrary, then theabove equation includes all solutions of the system, and it iscustomary to call it the general solution.
Any set of solutions x(1), ..., x(n) of the homogeneus system that islinearly independent at each point in the interval α < t < βis saidto be a fundamental set of solutions for that interval.
Theorem 7.6
If x(1), ..., x(n) are solutions of the homogeneus system on theinterval α < t < β,
then in this interval W [x(1), ..., x(n)] given by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
If the constants c1, ..., cn are thought of as arbitrary, then theabove equation includes all solutions of the system, and it iscustomary to call it the general solution.
Any set of solutions x(1), ..., x(n) of the homogeneus system that islinearly independent at each point in the interval α < t < βis saidto be a fundamental set of solutions for that interval.
Theorem 7.6
If x(1), ..., x(n) are solutions of the homogeneus system on theinterval α < t < β, then in this interval W [x(1), ..., x(n)] given by
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
W [x(1), x(2) . . . x(n)] =
∣∣∣∣∣∣∣∣∣∣x(1)1 x
(2)1 . . . x
(n)1
x(1)2 x
(2)2 . . . x
(n)2
...... . . .
...
x(1)n x
(2)n . . . x
(n)n
∣∣∣∣∣∣∣∣∣∣either is identically zero or else never vanishes.To prove this theorem is necessary to establish that
dW
dt= [p11 + p22 + ...+ pnn]W
Hence
W (t) = ce∫[p11+p22+...+pnn]dt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
W [x(1), x(2) . . . x(n)] =
∣∣∣∣∣∣∣∣∣∣x(1)1 x
(2)1 . . . x
(n)1
x(1)2 x
(2)2 . . . x
(n)2
...... . . .
...
x(1)n x
(2)n . . . x
(n)n
∣∣∣∣∣∣∣∣∣∣
either is identically zero or else never vanishes.To prove this theorem is necessary to establish that
dW
dt= [p11 + p22 + ...+ pnn]W
Hence
W (t) = ce∫[p11+p22+...+pnn]dt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
W [x(1), x(2) . . . x(n)] =
∣∣∣∣∣∣∣∣∣∣x(1)1 x
(2)1 . . . x
(n)1
x(1)2 x
(2)2 . . . x
(n)2
...... . . .
...
x(1)n x
(2)n . . . x
(n)n
∣∣∣∣∣∣∣∣∣∣either is identically zero or else never vanishes.
To prove this theorem is necessary to establish that
dW
dt= [p11 + p22 + ...+ pnn]W
Hence
W (t) = ce∫[p11+p22+...+pnn]dt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
W [x(1), x(2) . . . x(n)] =
∣∣∣∣∣∣∣∣∣∣x(1)1 x
(2)1 . . . x
(n)1
x(1)2 x
(2)2 . . . x
(n)2
...... . . .
...
x(1)n x
(2)n . . . x
(n)n
∣∣∣∣∣∣∣∣∣∣either is identically zero or else never vanishes.To prove this theorem is necessary to establish that
dW
dt= [p11 + p22 + ...+ pnn]W
Hence
W (t) = ce∫[p11+p22+...+pnn]dt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
W [x(1), x(2) . . . x(n)] =
∣∣∣∣∣∣∣∣∣∣x(1)1 x
(2)1 . . . x
(n)1
x(1)2 x
(2)2 . . . x
(n)2
...... . . .
...
x(1)n x
(2)n . . . x
(n)n
∣∣∣∣∣∣∣∣∣∣either is identically zero or else never vanishes.To prove this theorem is necessary to establish that
dW
dt= [p11 + p22 + ...+ pnn]W
Hence
W (t) = ce∫[p11+p22+...+pnn]dt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
W [x(1), x(2) . . . x(n)] =
∣∣∣∣∣∣∣∣∣∣x(1)1 x
(2)1 . . . x
(n)1
x(1)2 x
(2)2 . . . x
(n)2
...... . . .
...
x(1)n x
(2)n . . . x
(n)n
∣∣∣∣∣∣∣∣∣∣either is identically zero or else never vanishes.To prove this theorem is necessary to establish that
dW
dt= [p11 + p22 + ...+ pnn]W
Hence
W (t) = ce∫[p11+p22+...+pnn]dt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
W [x(1), x(2) . . . x(n)] =
∣∣∣∣∣∣∣∣∣∣x(1)1 x
(2)1 . . . x
(n)1
x(1)2 x
(2)2 . . . x
(n)2
...... . . .
...
x(1)n x
(2)n . . . x
(n)n
∣∣∣∣∣∣∣∣∣∣either is identically zero or else never vanishes.To prove this theorem is necessary to establish that
dW
dt= [p11 + p22 + ...+ pnn]W
Hence
W (t) = ce∫[p11+p22+...+pnn]dt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.7
Let x(1), ..., x(n) be the solutions of the homogeneus system thatsatisfy the initial conditions x(1)(t0) = e(1), x(1)(t0) = e(2),..., x(n)(t0) = e(n), respectively, where t0 is any point inα < t < βand
e(1) =
10...0
e(2) =
01...0
· · · e(n) =
00...1
Then, x(1), ..., x(n) form a fundamental set of solutions of thehomogeneous system.
Finally in the case that the solution is complex-valued, we have thefollowing result.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.7
Let x(1), ..., x(n) be the solutions of the homogeneus system thatsatisfy the initial conditions x(1)(t0) = e(1), x(1)(t0) = e(2),..., x(n)(t0) = e(n), respectively, where t0 is any point inα < t < βand
e(1) =
10...0
e(2) =
01...0
· · · e(n) =
00...1
Then, x(1), ..., x(n) form a fundamental set of solutions of thehomogeneous system.
Finally in the case that the solution is complex-valued, we have thefollowing result.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.7
Let x(1), ..., x(n)
be the solutions of the homogeneus system thatsatisfy the initial conditions x(1)(t0) = e(1), x(1)(t0) = e(2),..., x(n)(t0) = e(n), respectively, where t0 is any point inα < t < βand
e(1) =
10...0
e(2) =
01...0
· · · e(n) =
00...1
Then, x(1), ..., x(n) form a fundamental set of solutions of thehomogeneous system.
Finally in the case that the solution is complex-valued, we have thefollowing result.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.7
Let x(1), ..., x(n) be the solutions of the homogeneus system
thatsatisfy the initial conditions x(1)(t0) = e(1), x(1)(t0) = e(2),..., x(n)(t0) = e(n), respectively, where t0 is any point inα < t < βand
e(1) =
10...0
e(2) =
01...0
· · · e(n) =
00...1
Then, x(1), ..., x(n) form a fundamental set of solutions of thehomogeneous system.
Finally in the case that the solution is complex-valued, we have thefollowing result.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.7
Let x(1), ..., x(n) be the solutions of the homogeneus system thatsatisfy the initial conditions x(1)(t0) = e(1),
x(1)(t0) = e(2),..., x(n)(t0) = e(n), respectively, where t0 is any point inα < t < βand
e(1) =
10...0
e(2) =
01...0
· · · e(n) =
00...1
Then, x(1), ..., x(n) form a fundamental set of solutions of thehomogeneous system.
Finally in the case that the solution is complex-valued, we have thefollowing result.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.7
Let x(1), ..., x(n) be the solutions of the homogeneus system thatsatisfy the initial conditions x(1)(t0) = e(1), x(1)(t0) = e(2),...,
x(n)(t0) = e(n), respectively, where t0 is any point inα < t < βand
e(1) =
10...0
e(2) =
01...0
· · · e(n) =
00...1
Then, x(1), ..., x(n) form a fundamental set of solutions of thehomogeneous system.
Finally in the case that the solution is complex-valued, we have thefollowing result.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.7
Let x(1), ..., x(n) be the solutions of the homogeneus system thatsatisfy the initial conditions x(1)(t0) = e(1), x(1)(t0) = e(2),..., x(n)(t0) = e(n), respectively,
where t0 is any point inα < t < βand
e(1) =
10...0
e(2) =
01...0
· · · e(n) =
00...1
Then, x(1), ..., x(n) form a fundamental set of solutions of thehomogeneous system.
Finally in the case that the solution is complex-valued, we have thefollowing result.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.7
Let x(1), ..., x(n) be the solutions of the homogeneus system thatsatisfy the initial conditions x(1)(t0) = e(1), x(1)(t0) = e(2),..., x(n)(t0) = e(n), respectively, where t0 is any point inα < t < βand
e(1) =
10...0
e(2) =
01...0
· · · e(n) =
00...1
Then, x(1), ..., x(n) form a fundamental set of solutions of thehomogeneous system.
Finally in the case that the solution is complex-valued, we have thefollowing result.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.7
Let x(1), ..., x(n) be the solutions of the homogeneus system thatsatisfy the initial conditions x(1)(t0) = e(1), x(1)(t0) = e(2),..., x(n)(t0) = e(n), respectively, where t0 is any point inα < t < βand
e(1) =
10...0
e(2) =
01...0
· · · e(n) =
00...1
Then, x(1), ..., x(n) form a fundamental set of solutions of thehomogeneous system.
Finally in the case that the solution is complex-valued, we have thefollowing result.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.7
Let x(1), ..., x(n) be the solutions of the homogeneus system thatsatisfy the initial conditions x(1)(t0) = e(1), x(1)(t0) = e(2),..., x(n)(t0) = e(n), respectively, where t0 is any point inα < t < βand
e(1) =
10...0
e(2) =
01...0
· · · e(n) =
00...1
Then, x(1), ..., x(n) form a fundamental set of solutions of thehomogeneous system.
Finally in the case that the solution is complex-valued, we have thefollowing result.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.7
Let x(1), ..., x(n) be the solutions of the homogeneus system thatsatisfy the initial conditions x(1)(t0) = e(1), x(1)(t0) = e(2),..., x(n)(t0) = e(n), respectively, where t0 is any point inα < t < βand
e(1) =
10...0
e(2) =
01...0
· · ·
e(n) =
00...1
Then, x(1), ..., x(n) form a fundamental set of solutions of thehomogeneous system.
Finally in the case that the solution is complex-valued, we have thefollowing result.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.7
Let x(1), ..., x(n) be the solutions of the homogeneus system thatsatisfy the initial conditions x(1)(t0) = e(1), x(1)(t0) = e(2),..., x(n)(t0) = e(n), respectively, where t0 is any point inα < t < βand
e(1) =
10...0
e(2) =
01...0
· · · e(n) =
00...1
Then, x(1), ..., x(n) form a fundamental set of solutions of thehomogeneous system.
Finally in the case that the solution is complex-valued, we have thefollowing result.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.7
Let x(1), ..., x(n) be the solutions of the homogeneus system thatsatisfy the initial conditions x(1)(t0) = e(1), x(1)(t0) = e(2),..., x(n)(t0) = e(n), respectively, where t0 is any point inα < t < βand
e(1) =
10...0
e(2) =
01...0
· · · e(n) =
00...1
Then,
x(1), ..., x(n) form a fundamental set of solutions of thehomogeneous system.
Finally in the case that the solution is complex-valued, we have thefollowing result.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.7
Let x(1), ..., x(n) be the solutions of the homogeneus system thatsatisfy the initial conditions x(1)(t0) = e(1), x(1)(t0) = e(2),..., x(n)(t0) = e(n), respectively, where t0 is any point inα < t < βand
e(1) =
10...0
e(2) =
01...0
· · · e(n) =
00...1
Then, x(1), ..., x(n)
form a fundamental set of solutions of thehomogeneous system.
Finally in the case that the solution is complex-valued, we have thefollowing result.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.7
Let x(1), ..., x(n) be the solutions of the homogeneus system thatsatisfy the initial conditions x(1)(t0) = e(1), x(1)(t0) = e(2),..., x(n)(t0) = e(n), respectively, where t0 is any point inα < t < βand
e(1) =
10...0
e(2) =
01...0
· · · e(n) =
00...1
Then, x(1), ..., x(n) form a fundamental set of solutions of thehomogeneous system.
Finally in the case that the solution is complex-valued, we have thefollowing result.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.7
Let x(1), ..., x(n) be the solutions of the homogeneus system thatsatisfy the initial conditions x(1)(t0) = e(1), x(1)(t0) = e(2),..., x(n)(t0) = e(n), respectively, where t0 is any point inα < t < βand
e(1) =
10...0
e(2) =
01...0
· · · e(n) =
00...1
Then, x(1), ..., x(n) form a fundamental set of solutions of thehomogeneous system.
Finally in the case that the solution is complex-valued, we have thefollowing result.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.8
Consider the homogeneous system
X′ = P(t)X
where each element of P is a real-valued continuous function. Ifx = u(t) + i v(t) is a complex-valued solution, then its real partu(t) and its imaginary part v(t) are also solutions of this equation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.8
Consider the homogeneous system
X′ = P(t)X
where each element of P is a real-valued continuous function. Ifx = u(t) + i v(t) is a complex-valued solution, then its real partu(t) and its imaginary part v(t) are also solutions of this equation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.8
Consider the homogeneous system
X′ = P(t)X
where each element of P is a real-valued continuous function. Ifx = u(t) + i v(t) is a complex-valued solution, then its real partu(t) and its imaginary part v(t) are also solutions of this equation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.8
Consider the homogeneous system
X′ = P(t)X
where each element of P is a real-valued continuous function. Ifx = u(t) + i v(t) is a complex-valued solution, then its real partu(t) and its imaginary part v(t) are also solutions of this equation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.8
Consider the homogeneous system
X′ = P(t)X
where each element of P is a real-valued continuous function.
Ifx = u(t) + i v(t) is a complex-valued solution, then its real partu(t) and its imaginary part v(t) are also solutions of this equation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.8
Consider the homogeneous system
X′ = P(t)X
where each element of P is a real-valued continuous function. Ifx = u(t) + i v(t) is a complex-valued solution,
then its real partu(t) and its imaginary part v(t) are also solutions of this equation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.8
Consider the homogeneous system
X′ = P(t)X
where each element of P is a real-valued continuous function. Ifx = u(t) + i v(t) is a complex-valued solution, then its real partu(t) and
its imaginary part v(t) are also solutions of this equation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.8
Consider the homogeneous system
X′ = P(t)X
where each element of P is a real-valued continuous function. Ifx = u(t) + i v(t) is a complex-valued solution, then its real partu(t) and its imaginary part v(t)
are also solutions of this equation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
IntroductionSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsSystems of Linear Algebraic Equations; Linear Independence, Eigenvalues, EigenvectorsBasic Theory of Systems of First Order Linear Equations
Basic Theory of Systems of First Order LinearEquations
Theorem 7.8
Consider the homogeneous system
X′ = P(t)X
where each element of P is a real-valued continuous function. Ifx = u(t) + i v(t) is a complex-valued solution, then its real partu(t) and its imaginary part v(t) are also solutions of this equation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
We will concentrate most of our attention on systems ofhomogeneous linear equations with constant coefficients
x′ = Ax
where A is a constant n × n matrix. Unless stated otherwise, wewill assume further that all the elements of A are real (rather thancomplex) numbers.
The case n = 2 is particularly important and lends itself tovisualization in the x1x2− plane, called the phase plane. Byevaluating Ax at a large number of points and plotting theresulting vectors, we obtain a direction field of tangent vectors tosolutions of the system of differential equations.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
We will concentrate most of our attention on systems ofhomogeneous linear equations with constant coefficients
x′ = Ax
where A is a constant n × n matrix. Unless stated otherwise, wewill assume further that all the elements of A are real (rather thancomplex) numbers.
The case n = 2 is particularly important and lends itself tovisualization in the x1x2− plane, called the phase plane. Byevaluating Ax at a large number of points and plotting theresulting vectors, we obtain a direction field of tangent vectors tosolutions of the system of differential equations.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
We will concentrate most of our attention on systems ofhomogeneous linear equations with constant coefficients
x′ = Ax
where A is a constant n × n matrix. Unless stated otherwise, wewill assume further that all the elements of A are real (rather thancomplex) numbers.
The case n = 2 is particularly important and lends itself tovisualization in the x1x2− plane, called the phase plane. Byevaluating Ax at a large number of points and plotting theresulting vectors, we obtain a direction field of tangent vectors tosolutions of the system of differential equations.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
We will concentrate most of our attention on systems ofhomogeneous linear equations with constant coefficients
x′ = Ax
where A is a constant n × n matrix.
Unless stated otherwise, wewill assume further that all the elements of A are real (rather thancomplex) numbers.
The case n = 2 is particularly important and lends itself tovisualization in the x1x2− plane, called the phase plane. Byevaluating Ax at a large number of points and plotting theresulting vectors, we obtain a direction field of tangent vectors tosolutions of the system of differential equations.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
We will concentrate most of our attention on systems ofhomogeneous linear equations with constant coefficients
x′ = Ax
where A is a constant n × n matrix. Unless stated otherwise,
wewill assume further that all the elements of A are real (rather thancomplex) numbers.
The case n = 2 is particularly important and lends itself tovisualization in the x1x2− plane, called the phase plane. Byevaluating Ax at a large number of points and plotting theresulting vectors, we obtain a direction field of tangent vectors tosolutions of the system of differential equations.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
We will concentrate most of our attention on systems ofhomogeneous linear equations with constant coefficients
x′ = Ax
where A is a constant n × n matrix. Unless stated otherwise, wewill assume further that all the elements of A
are real (rather thancomplex) numbers.
The case n = 2 is particularly important and lends itself tovisualization in the x1x2− plane, called the phase plane. Byevaluating Ax at a large number of points and plotting theresulting vectors, we obtain a direction field of tangent vectors tosolutions of the system of differential equations.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
We will concentrate most of our attention on systems ofhomogeneous linear equations with constant coefficients
x′ = Ax
where A is a constant n × n matrix. Unless stated otherwise, wewill assume further that all the elements of A are real (rather thancomplex) numbers.
The case n = 2 is particularly important and lends itself tovisualization in the x1x2− plane, called the phase plane. Byevaluating Ax at a large number of points and plotting theresulting vectors, we obtain a direction field of tangent vectors tosolutions of the system of differential equations.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
We will concentrate most of our attention on systems ofhomogeneous linear equations with constant coefficients
x′ = Ax
where A is a constant n × n matrix. Unless stated otherwise, wewill assume further that all the elements of A are real (rather thancomplex) numbers.
The case n = 2 is particularly important and lends itself tovisualization in the x1x2− plane, called the phase plane.
Byevaluating Ax at a large number of points and plotting theresulting vectors, we obtain a direction field of tangent vectors tosolutions of the system of differential equations.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
We will concentrate most of our attention on systems ofhomogeneous linear equations with constant coefficients
x′ = Ax
where A is a constant n × n matrix. Unless stated otherwise, wewill assume further that all the elements of A are real (rather thancomplex) numbers.
The case n = 2 is particularly important and lends itself tovisualization in the x1x2− plane, called the phase plane. Byevaluating Ax
at a large number of points and plotting theresulting vectors, we obtain a direction field of tangent vectors tosolutions of the system of differential equations.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
We will concentrate most of our attention on systems ofhomogeneous linear equations with constant coefficients
x′ = Ax
where A is a constant n × n matrix. Unless stated otherwise, wewill assume further that all the elements of A are real (rather thancomplex) numbers.
The case n = 2 is particularly important and lends itself tovisualization in the x1x2− plane, called the phase plane. Byevaluating Ax at a large number of points and
plotting theresulting vectors, we obtain a direction field of tangent vectors tosolutions of the system of differential equations.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
We will concentrate most of our attention on systems ofhomogeneous linear equations with constant coefficients
x′ = Ax
where A is a constant n × n matrix. Unless stated otherwise, wewill assume further that all the elements of A are real (rather thancomplex) numbers.
The case n = 2 is particularly important and lends itself tovisualization in the x1x2− plane, called the phase plane. Byevaluating Ax at a large number of points and plotting theresulting vectors,
we obtain a direction field of tangent vectors tosolutions of the system of differential equations.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
We will concentrate most of our attention on systems ofhomogeneous linear equations with constant coefficients
x′ = Ax
where A is a constant n × n matrix. Unless stated otherwise, wewill assume further that all the elements of A are real (rather thancomplex) numbers.
The case n = 2 is particularly important and lends itself tovisualization in the x1x2− plane, called the phase plane. Byevaluating Ax at a large number of points and plotting theresulting vectors, we obtain a direction field of tangent vectors to
solutions of the system of differential equations.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
We will concentrate most of our attention on systems ofhomogeneous linear equations with constant coefficients
x′ = Ax
where A is a constant n × n matrix. Unless stated otherwise, wewill assume further that all the elements of A are real (rather thancomplex) numbers.
The case n = 2 is particularly important and lends itself tovisualization in the x1x2− plane, called the phase plane. Byevaluating Ax at a large number of points and plotting theresulting vectors, we obtain a direction field of tangent vectors tosolutions of the system of differential equations.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
A qualitative understanding of the behavior of solutions can usuallybe gained from a direction field. More precise information resultsfrom including in the plot some solution curves, or trajectories. Aplot that shows a representative sample of trajectories for a givensystem is called a phase portrait .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
A qualitative understanding of the behavior of solutions
can usuallybe gained from a direction field. More precise information resultsfrom including in the plot some solution curves, or trajectories. Aplot that shows a representative sample of trajectories for a givensystem is called a phase portrait .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
A qualitative understanding of the behavior of solutions can usuallybe gained from a direction field.
More precise information resultsfrom including in the plot some solution curves, or trajectories. Aplot that shows a representative sample of trajectories for a givensystem is called a phase portrait .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
A qualitative understanding of the behavior of solutions can usuallybe gained from a direction field. More precise information results
from including in the plot some solution curves, or trajectories. Aplot that shows a representative sample of trajectories for a givensystem is called a phase portrait .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
A qualitative understanding of the behavior of solutions can usuallybe gained from a direction field. More precise information resultsfrom including in the plot some solution curves, or trajectories.
Aplot that shows a representative sample of trajectories for a givensystem is called a phase portrait .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
A qualitative understanding of the behavior of solutions can usuallybe gained from a direction field. More precise information resultsfrom including in the plot some solution curves, or trajectories. Aplot that shows a representative sample of trajectories for a givensystem is called a phase portrait .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
A qualitative understanding of the behavior of solutions can usuallybe gained from a direction field. More precise information resultsfrom including in the plot some solution curves, or trajectories. Aplot that shows a representative sample of trajectories for a givensystem is called a phase portrait .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Now, for the system
x′ = Ax
we look for solutions of the form
x = veλt
where the expon entλ and the vector v are to be determined.Substituting x in the system gives
λveλt = Aveλt
(A− λI) v = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Now, for the system
x′ = Ax
we look for solutions of the form
x = veλt
where the expon entλ and the vector v are to be determined.Substituting x in the system gives
λveλt = Aveλt
(A− λI) v = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Now, for the system
x′ = Ax
we look for solutions of the form
x = veλt
where the expon entλ and the vector v are to be determined.Substituting x in the system gives
λveλt = Aveλt
(A− λI) v = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Now, for the system
x′ = Ax
we look for solutions of the form
x = veλt
where the expon entλ and the vector v are to be determined.Substituting x in the system gives
λveλt = Aveλt
(A− λI) v = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Now, for the system
x′ = Ax
we look for solutions of the form
x = veλt
where the expon entλ and the vector v are to be determined.Substituting x in the system gives
λveλt = Aveλt
(A− λI) v = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Now, for the system
x′ = Ax
we look for solutions of the form
x = veλt
where the expon entλ and the vector v are to be determined.
Substituting x in the system gives
λveλt = Aveλt
(A− λI) v = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Now, for the system
x′ = Ax
we look for solutions of the form
x = veλt
where the expon entλ and the vector v are to be determined.Substituting x in the system gives
λveλt = Aveλt
(A− λI) v = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Now, for the system
x′ = Ax
we look for solutions of the form
x = veλt
where the expon entλ and the vector v are to be determined.Substituting x in the system gives
λveλt = Aveλt
(A− λI) v = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Now, for the system
x′ = Ax
we look for solutions of the form
x = veλt
where the expon entλ and the vector v are to be determined.Substituting x in the system gives
λveλt = Aveλt
(A− λI) v = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Thus, to solve the system of differential equations, we must solvethe above system of algebraic equations. That is, we need to findthe eigenvalues and eigenvectors of the matrix A.
If we assume that A is a real-valued matrix, then we must considerthe following possibilities for the eigenvalues of A:
1. All eigenvalues are real and different from each other.
2. Some eigenvalues occur in complex conjugate pairs.
3. Some eigenvalues, either real or complex, are repeated. If the neigenvalues are all real and different, as in the
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Thus, to solve the system of differential equations,
we must solvethe above system of algebraic equations. That is, we need to findthe eigenvalues and eigenvectors of the matrix A.
If we assume that A is a real-valued matrix, then we must considerthe following possibilities for the eigenvalues of A:
1. All eigenvalues are real and different from each other.
2. Some eigenvalues occur in complex conjugate pairs.
3. Some eigenvalues, either real or complex, are repeated. If the neigenvalues are all real and different, as in the
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Thus, to solve the system of differential equations, we must solvethe above system of algebraic equations.
That is, we need to findthe eigenvalues and eigenvectors of the matrix A.
If we assume that A is a real-valued matrix, then we must considerthe following possibilities for the eigenvalues of A:
1. All eigenvalues are real and different from each other.
2. Some eigenvalues occur in complex conjugate pairs.
3. Some eigenvalues, either real or complex, are repeated. If the neigenvalues are all real and different, as in the
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Thus, to solve the system of differential equations, we must solvethe above system of algebraic equations. That is, we need to find
the eigenvalues and eigenvectors of the matrix A.
If we assume that A is a real-valued matrix, then we must considerthe following possibilities for the eigenvalues of A:
1. All eigenvalues are real and different from each other.
2. Some eigenvalues occur in complex conjugate pairs.
3. Some eigenvalues, either real or complex, are repeated. If the neigenvalues are all real and different, as in the
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Thus, to solve the system of differential equations, we must solvethe above system of algebraic equations. That is, we need to findthe eigenvalues and eigenvectors of the matrix A.
If we assume that A is a real-valued matrix, then we must considerthe following possibilities for the eigenvalues of A:
1. All eigenvalues are real and different from each other.
2. Some eigenvalues occur in complex conjugate pairs.
3. Some eigenvalues, either real or complex, are repeated. If the neigenvalues are all real and different, as in the
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Thus, to solve the system of differential equations, we must solvethe above system of algebraic equations. That is, we need to findthe eigenvalues and eigenvectors of the matrix A.
If we assume that A is a real-valued matrix,
then we must considerthe following possibilities for the eigenvalues of A:
1. All eigenvalues are real and different from each other.
2. Some eigenvalues occur in complex conjugate pairs.
3. Some eigenvalues, either real or complex, are repeated. If the neigenvalues are all real and different, as in the
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Thus, to solve the system of differential equations, we must solvethe above system of algebraic equations. That is, we need to findthe eigenvalues and eigenvectors of the matrix A.
If we assume that A is a real-valued matrix, then we must considerthe following possibilities for the eigenvalues of A:
1. All eigenvalues are real and different from each other.
2. Some eigenvalues occur in complex conjugate pairs.
3. Some eigenvalues, either real or complex, are repeated. If the neigenvalues are all real and different, as in the
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Thus, to solve the system of differential equations, we must solvethe above system of algebraic equations. That is, we need to findthe eigenvalues and eigenvectors of the matrix A.
If we assume that A is a real-valued matrix, then we must considerthe following possibilities for the eigenvalues of A:
1. All eigenvalues are real and different from each other.
2. Some eigenvalues occur in complex conjugate pairs.
3. Some eigenvalues, either real or complex, are repeated. If the neigenvalues are all real and different, as in the
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Thus, to solve the system of differential equations, we must solvethe above system of algebraic equations. That is, we need to findthe eigenvalues and eigenvectors of the matrix A.
If we assume that A is a real-valued matrix, then we must considerthe following possibilities for the eigenvalues of A:
1. All eigenvalues are real and different from each other.
2. Some eigenvalues occur in complex conjugate pairs.
3. Some eigenvalues, either real or complex, are repeated. If the neigenvalues are all real and different, as in the
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Thus, to solve the system of differential equations, we must solvethe above system of algebraic equations. That is, we need to findthe eigenvalues and eigenvectors of the matrix A.
If we assume that A is a real-valued matrix, then we must considerthe following possibilities for the eigenvalues of A:
1. All eigenvalues are real and different from each other.
2. Some eigenvalues occur in complex conjugate pairs.
3. Some eigenvalues, either real or complex, are repeated. If the neigenvalues are all real and different, as in the
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Example 7.14
Consider the system
x′ = Ax =
(1 14 1
)x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Example 7.14
Consider the system
x′ = Ax =
(1 14 1
)x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Example 7.14
Consider the system
x′ = Ax =
(1 14 1
)x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Example 7.14
Consider the system
x′ = Ax =
(1 14 1
)x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Example 7.14
Consider the system
x′ = Ax =
(1 14 1
)x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Example 7.14
Consider the system
x′ = Ax =
(1 14 1
)x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Example 7.14
Consider the system
x′ = Ax =
(1 14 1
)x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
|A− λI| =
∣∣∣∣1− λ 14 1− λ
∣∣∣∣ = 0
(1− λ)2 − 4 = 0 =⇒
(λ2 − 2λ− 3 = (λ− 3)(λ+ 1) = 0 =⇒
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
|A− λI| =
∣∣∣∣1− λ 14 1− λ
∣∣∣∣ = 0
(1− λ)2 − 4 = 0 =⇒
(λ2 − 2λ− 3 = (λ− 3)(λ+ 1) = 0 =⇒
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
|A− λI| =
∣∣∣∣1− λ 14 1− λ
∣∣∣∣ = 0
(1− λ)2 − 4 = 0 =⇒
(λ2 − 2λ− 3 = (λ− 3)(λ+ 1) = 0 =⇒
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
|A− λI| =
∣∣∣∣1− λ 14 1− λ
∣∣∣∣ = 0
(1− λ)2 − 4 = 0 =⇒
(λ2 − 2λ− 3 = (λ− 3)(λ+ 1) = 0 =⇒
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
|A− λI| =
∣∣∣∣1− λ 14 1− λ
∣∣∣∣ = 0
(1− λ)2 − 4 = 0 =⇒
(λ2 − 2λ− 3 = (λ− 3)(λ+ 1) = 0 =⇒
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
λ1 = 3, λ2 = −1
If λ1 = 3, then the system reduces to the single equation
−2v1 + v2 = 0, =⇒ v2 = 2v1
and a corresponding eigenvector is
v(1) =
(12
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
λ1 = 3, λ2 = −1
If λ1 = 3, then the system reduces to the single equation
−2v1 + v2 = 0, =⇒ v2 = 2v1
and a corresponding eigenvector is
v(1) =
(12
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
λ1 = 3, λ2 = −1
If λ1 = 3, then the system reduces to the single equation
−2v1 + v2 = 0, =⇒ v2 = 2v1
and a corresponding eigenvector is
v(1) =
(12
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
λ1 = 3, λ2 = −1
If λ1 = 3, then the system reduces to the single equation
−2v1 + v2 = 0, =⇒ v2 = 2v1
and a corresponding eigenvector is
v(1) =
(12
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
λ1 = 3, λ2 = −1
If λ1 = 3, then the system reduces to the single equation
−2v1 + v2 = 0, =⇒ v2 = 2v1
and a corresponding eigenvector is
v(1) =
(12
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
λ1 = 3, λ2 = −1
If λ1 = 3, then the system reduces to the single equation
−2v1 + v2 = 0, =⇒ v2 = 2v1
and a corresponding eigenvector is
v(1) =
(12
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Similarly, corresponding to λ2 = −1, we find that a correspondingeigenvector is
v(2) =
(1
− 2
)The corresponding solutions of the differential equation are
x(1) =
(12
)e3t ; x(2) =
(1
− 2
)e−t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Similarly, corresponding to λ2 = −1, we find that a correspondingeigenvector is
v(2) =
(1
− 2
)The corresponding solutions of the differential equation are
x(1) =
(12
)e3t ; x(2) =
(1
− 2
)e−t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Similarly, corresponding to λ2 = −1, we find that a correspondingeigenvector is
v(2) =
(1
− 2
)
The corresponding solutions of the differential equation are
x(1) =
(12
)e3t ; x(2) =
(1
− 2
)e−t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Similarly, corresponding to λ2 = −1, we find that a correspondingeigenvector is
v(2) =
(1
− 2
)The corresponding solutions of the differential equation are
x(1) =
(12
)e3t ; x(2) =
(1
− 2
)e−t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Similarly, corresponding to λ2 = −1, we find that a correspondingeigenvector is
v(2) =
(1
− 2
)The corresponding solutions of the differential equation are
x(1) =
(12
)e3t ;
x(2) =
(1
− 2
)e−t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
Similarly, corresponding to λ2 = −1, we find that a correspondingeigenvector is
v(2) =
(1
− 2
)The corresponding solutions of the differential equation are
x(1) =
(12
)e3t ; x(2) =
(1
− 2
)e−t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
The Wronskian of these solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e3t e−t
2e3t −2e−t
∣∣∣∣ = − 4e2t 6= 0
Hence the solutions x(1) and x(2) form a fundamental set, and thegeneral solution of the system is
x = c1x(1) + c2x(2) = c1
(12
)e3t + c2
(1
− 2
)e−t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
The Wronskian of these solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e3t e−t
2e3t −2e−t
∣∣∣∣ = − 4e2t 6= 0
Hence the solutions x(1) and x(2) form a fundamental set, and thegeneral solution of the system is
x = c1x(1) + c2x(2) = c1
(12
)e3t + c2
(1
− 2
)e−t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
The Wronskian of these solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e3t e−t
2e3t −2e−t
∣∣∣∣ = − 4e2t 6= 0
Hence the solutions x(1) and x(2) form a fundamental set, and thegeneral solution of the system is
x = c1x(1) + c2x(2) = c1
(12
)e3t + c2
(1
− 2
)e−t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
The Wronskian of these solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e3t e−t
2e3t −2e−t
∣∣∣∣ =
− 4e2t 6= 0
Hence the solutions x(1) and x(2) form a fundamental set, and thegeneral solution of the system is
x = c1x(1) + c2x(2) = c1
(12
)e3t + c2
(1
− 2
)e−t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
The Wronskian of these solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e3t e−t
2e3t −2e−t
∣∣∣∣ = − 4e2t 6= 0
Hence the solutions x(1) and x(2) form a fundamental set, and thegeneral solution of the system is
x = c1x(1) + c2x(2) = c1
(12
)e3t + c2
(1
− 2
)e−t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
The Wronskian of these solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e3t e−t
2e3t −2e−t
∣∣∣∣ = − 4e2t 6= 0
Hence the solutions x(1) and x(2) form a fundamental set, and thegeneral solution of the system is
x = c1x(1) + c2x(2) = c1
(12
)e3t + c2
(1
− 2
)e−t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
The Wronskian of these solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e3t e−t
2e3t −2e−t
∣∣∣∣ = − 4e2t 6= 0
Hence the solutions x(1) and x(2) form a fundamental set, and thegeneral solution of the system is
x =
c1x(1) + c2x(2) = c1
(12
)e3t + c2
(1
− 2
)e−t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
The Wronskian of these solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e3t e−t
2e3t −2e−t
∣∣∣∣ = − 4e2t 6= 0
Hence the solutions x(1) and x(2) form a fundamental set, and thegeneral solution of the system is
x = c1x(1) + c2x(2) =
c1
(12
)e3t + c2
(1
− 2
)e−t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
The Wronskian of these solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e3t e−t
2e3t −2e−t
∣∣∣∣ = − 4e2t 6= 0
Hence the solutions x(1) and x(2) form a fundamental set, and thegeneral solution of the system is
x = c1x(1) + c2x(2) = c1
(12
)e3t +
c2
(1
− 2
)e−t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Homogeneous Linear Systems with ConstantCoefficients
The Wronskian of these solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e3t e−t
2e3t −2e−t
∣∣∣∣ = − 4e2t 6= 0
Hence the solutions x(1) and x(2) form a fundamental set, and thegeneral solution of the system is
x = c1x(1) + c2x(2) = c1
(12
)e3t + c2
(1
− 2
)e−t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued.
If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt ,
then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand
v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector
of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex,
we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate.
Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν;
v±k = a± i b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
In this section we consider again a system of n linear homogeneousequations with constant coefficients
X′ = AX
where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.
In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that
λ±k = µ± i ν; v±k = a± i b
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors de lamatrix a, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors de lamatrix a, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors de lamatrix a, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors de lamatrix a, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors de lamatrix a, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t =
eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors de lamatrix a, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors de lamatrix a, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors de lamatrix a, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)
X2(t) =1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
are two complex-conjugate eigenvalues and eigenvectors de lamatrix a, then
X±(t) = e(µ±i ν)t (a± i b)
are complex-valued solutions, but taking in account that
e(µ±i ν)t = eµt (cos(νt)± i sin(νt))
and the principle of superposition, then we have that
X1(t) =1
2
(X+(t) + X−(t)
)X2(t) =
1
2i
(X+(t)− X−(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
are two (real) solutions !!!
X1(t) = eµt (acos(νt)− bsin(νt))
X2(t) = eµt (acos(νt) + bsin(νt))
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
are two (real) solutions !!!
X1(t) = eµt (acos(νt)− bsin(νt))
X2(t) = eµt (acos(νt) + bsin(νt))
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
are two (real) solutions !!!
X1(t) = eµt (acos(νt)− bsin(νt))
X2(t) = eµt (acos(νt) + bsin(νt))
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
are two (real) solutions !!!
X1(t) = eµt (acos(νt)− bsin(νt))
X2(t) = eµt (acos(νt) + bsin(νt))
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Example 7.17
Solve the following ODE
x′ = Ax =
3 1 10 2 10 −1 2
x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Example 7.17
Solve the following ODE
x′ = Ax =
3 1 10 2 10 −1 2
x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Example 7.17
Solve the following ODE
x′ = Ax =
3 1 10 2 10 −1 2
x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Example 7.17
Solve the following ODE
x′ = Ax =
3 1 10 2 10 −1 2
x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Example 7.17
Solve the following ODE
x′ = Ax =
3 1 10 2 10 −1 2
x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Example 7.17
Solve the following ODE
x′ = Ax =
3 1 10 2 10 −1 2
x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Example 7.17
Solve the following ODE
x′ = Ax =
3 1 10 2 10 −1 2
x
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
|A− λI| =
∣∣∣∣∣∣3− λ 1 1
0 2− λ 10 −1 2− λ
∣∣∣∣∣∣ = 0
(3− λ)
∣∣∣∣2− λ 1−1 2− λ
∣∣∣∣− (1)
∣∣∣∣0 10 2− λ
∣∣∣∣+ (1)
∣∣∣∣1 2− λ1 1
∣∣∣∣ =⇒
(3− λ)(λ2 − 4λ+ 6) = 0 =⇒
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
|A− λI| =
∣∣∣∣∣∣3− λ 1 1
0 2− λ 10 −1 2− λ
∣∣∣∣∣∣ = 0
(3− λ)
∣∣∣∣2− λ 1−1 2− λ
∣∣∣∣− (1)
∣∣∣∣0 10 2− λ
∣∣∣∣+ (1)
∣∣∣∣1 2− λ1 1
∣∣∣∣ =⇒
(3− λ)(λ2 − 4λ+ 6) = 0 =⇒
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
|A− λI| =
∣∣∣∣∣∣3− λ 1 1
0 2− λ 10 −1 2− λ
∣∣∣∣∣∣ = 0
(3− λ)
∣∣∣∣2− λ 1−1 2− λ
∣∣∣∣− (1)
∣∣∣∣0 10 2− λ
∣∣∣∣+ (1)
∣∣∣∣1 2− λ1 1
∣∣∣∣ =⇒
(3− λ)(λ2 − 4λ+ 6) = 0 =⇒
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
|A− λI| =
∣∣∣∣∣∣3− λ 1 1
0 2− λ 10 −1 2− λ
∣∣∣∣∣∣ = 0
(3− λ)
∣∣∣∣2− λ 1−1 2− λ
∣∣∣∣−
(1)
∣∣∣∣0 10 2− λ
∣∣∣∣+ (1)
∣∣∣∣1 2− λ1 1
∣∣∣∣ =⇒
(3− λ)(λ2 − 4λ+ 6) = 0 =⇒
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
|A− λI| =
∣∣∣∣∣∣3− λ 1 1
0 2− λ 10 −1 2− λ
∣∣∣∣∣∣ = 0
(3− λ)
∣∣∣∣2− λ 1−1 2− λ
∣∣∣∣− (1)
∣∣∣∣0 10 2− λ
∣∣∣∣+ (1)
∣∣∣∣1 2− λ1 1
∣∣∣∣ =⇒
(3− λ)(λ2 − 4λ+ 6) = 0 =⇒
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
|A− λI| =
∣∣∣∣∣∣3− λ 1 1
0 2− λ 10 −1 2− λ
∣∣∣∣∣∣ = 0
(3− λ)
∣∣∣∣2− λ 1−1 2− λ
∣∣∣∣− (1)
∣∣∣∣0 10 2− λ
∣∣∣∣+ (1)
∣∣∣∣1 2− λ1 1
∣∣∣∣ =⇒
(3− λ)(λ2 − 4λ+ 6) = 0 =⇒
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
λ1 = 2, λ2,3 =4±
√16− (4)(5)
2= 2± i
If λ1 = 3, then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
0 1 10 −1 10 −1 −1
v1v2v3
=
0 1 10 0 20 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
λ1 = 2,
λ2,3 =4±
√16− (4)(5)
2= 2± i
If λ1 = 3, then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
0 1 10 −1 10 −1 −1
v1v2v3
=
0 1 10 0 20 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
λ1 = 2, λ2,3 =4±
√16− (4)(5)
2=
2± i
If λ1 = 3, then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
0 1 10 −1 10 −1 −1
v1v2v3
=
0 1 10 0 20 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
λ1 = 2, λ2,3 =4±
√16− (4)(5)
2= 2± i
If λ1 = 3, then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
0 1 10 −1 10 −1 −1
v1v2v3
=
0 1 10 0 20 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
λ1 = 2, λ2,3 =4±
√16− (4)(5)
2= 2± i
If λ1 = 3, then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
0 1 10 −1 10 −1 −1
v1v2v3
=
0 1 10 0 20 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
λ1 = 2, λ2,3 =4±
√16− (4)(5)
2= 2± i
If λ1 = 3, then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
0 1 10 −1 10 −1 −1
v1v2v3
=
0 1 10 0 20 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
λ1 = 2, λ2,3 =4±
√16− (4)(5)
2= 2± i
If λ1 = 3, then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
0 1 10 −1 10 −1 −1
v1v2v3
=
0 1 10 0 20 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
λ1 = 2, λ2,3 =4±
√16− (4)(5)
2= 2± i
If λ1 = 3, then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
0 1 10 −1 10 −1 −1
v1v2v3
=
0 1 10 0 20 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
λ1 = 2, λ2,3 =4±
√16− (4)(5)
2= 2± i
If λ1 = 3, then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
0 1 10 −1 10 −1 −1
v1v2v3
=
0 1 10 0 20 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
100
If λ2 = 2 + i , then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
100
If λ2 = 2 + i , then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
100
If λ2 = 2 + i , then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
100
If λ2 = 2 + i , then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
100
If λ2 = 2 + i , then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
100
If λ2 = 2 + i , then
(A− λ1I) v =
3− λ 1 10 2− λ 10 −1 2− λ
v1v2v3
=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
3− (2 + i) 1 10 2− (2 + i) 10 −1 2− (2 + i)
v1v2v3
=
1− i 1 10 −i 10 −1 −i
v1v2v3
=
1− i 1 10 −i 10 0 0
v1v2v3
=
1− i 0 1− i0 −i 10 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
3− (2 + i) 1 10 2− (2 + i) 10 −1 2− (2 + i)
v1v2v3
=
1− i 1 10 −i 10 −1 −i
v1v2v3
=
1− i 1 10 −i 10 0 0
v1v2v3
=
1− i 0 1− i0 −i 10 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
3− (2 + i) 1 10 2− (2 + i) 10 −1 2− (2 + i)
v1v2v3
=
1− i 1 10 −i 10 −1 −i
v1v2v3
=
1− i 1 10 −i 10 0 0
v1v2v3
=
1− i 0 1− i0 −i 10 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
3− (2 + i) 1 10 2− (2 + i) 10 −1 2− (2 + i)
v1v2v3
=
1− i 1 10 −i 10 −1 −i
v1v2v3
=
1− i 1 10 −i 10 0 0
v1v2v3
=
1− i 0 1− i0 −i 10 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
3− (2 + i) 1 10 2− (2 + i) 10 −1 2− (2 + i)
v1v2v3
=
1− i 1 10 −i 10 −1 −i
v1v2v3
=
1− i 1 10 −i 10 0 0
v1v2v3
=
1− i 0 1− i0 −i 10 0 0
v1v2v3
=
000
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+
i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ;
x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(2) =
10−1
+ i
010
The corresponding solutions of the differential equation are
x(1) =
100
e3t ; x(2) = e2t
10−1
cos(t)−
010
sin(t)
x(3) = e2t
10−1
cos(t) +
010
sin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2), x(3)](t) =
∣∣∣∣∣∣e3t e2tcos(t) e2tsin(t)0 −e2tsin(t) e2tcos(t)0 −e2tcos(t) −e2tsin(t)
∣∣∣∣∣∣ =
e3te2te2t
∣∣∣∣∣∣1 cos(t) sin(t)0 −sin(t) cos(t)0 −cos(t) −sin(t)
∣∣∣∣∣∣ =
e3te2te2t(sin2(t) + cos2(t)
)= e7t 6= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2), x(3)](t) =
∣∣∣∣∣∣e3t e2tcos(t) e2tsin(t)0 −e2tsin(t) e2tcos(t)0 −e2tcos(t) −e2tsin(t)
∣∣∣∣∣∣ =
e3te2te2t
∣∣∣∣∣∣1 cos(t) sin(t)0 −sin(t) cos(t)0 −cos(t) −sin(t)
∣∣∣∣∣∣ =
e3te2te2t(sin2(t) + cos2(t)
)= e7t 6= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2), x(3)](t) =
∣∣∣∣∣∣e3t e2tcos(t) e2tsin(t)0 −e2tsin(t) e2tcos(t)0 −e2tcos(t) −e2tsin(t)
∣∣∣∣∣∣ =
e3te2te2t
∣∣∣∣∣∣1 cos(t) sin(t)0 −sin(t) cos(t)0 −cos(t) −sin(t)
∣∣∣∣∣∣ =
e3te2te2t(sin2(t) + cos2(t)
)= e7t 6= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2), x(3)](t) =
∣∣∣∣∣∣e3t e2tcos(t) e2tsin(t)0 −e2tsin(t) e2tcos(t)0 −e2tcos(t) −e2tsin(t)
∣∣∣∣∣∣ =
e3te2te2t
∣∣∣∣∣∣1 cos(t) sin(t)0 −sin(t) cos(t)0 −cos(t) −sin(t)
∣∣∣∣∣∣ =
e3te2te2t(sin2(t) + cos2(t)
)= e7t 6= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2), x(3)](t) =
∣∣∣∣∣∣e3t e2tcos(t) e2tsin(t)0 −e2tsin(t) e2tcos(t)0 −e2tcos(t) −e2tsin(t)
∣∣∣∣∣∣ =
e3te2te2t
∣∣∣∣∣∣1 cos(t) sin(t)0 −sin(t) cos(t)0 −cos(t) −sin(t)
∣∣∣∣∣∣ =
e3te2te2t(sin2(t) + cos2(t)
)= e7t 6= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2), x(3)](t) =
∣∣∣∣∣∣e3t e2tcos(t) e2tsin(t)0 −e2tsin(t) e2tcos(t)0 −e2tcos(t) −e2tsin(t)
∣∣∣∣∣∣ =
e3te2te2t
∣∣∣∣∣∣1 cos(t) sin(t)0 −sin(t) cos(t)0 −cos(t) −sin(t)
∣∣∣∣∣∣ =
e3te2te2t(sin2(t) + cos2(t)
)=
e7t 6= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2), x(3)](t) =
∣∣∣∣∣∣e3t e2tcos(t) e2tsin(t)0 −e2tsin(t) e2tcos(t)0 −e2tcos(t) −e2tsin(t)
∣∣∣∣∣∣ =
e3te2te2t
∣∣∣∣∣∣1 cos(t) sin(t)0 −sin(t) cos(t)0 −cos(t) −sin(t)
∣∣∣∣∣∣ =
e3te2te2t(sin2(t) + cos2(t)
)= e7t
6= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2), x(3)](t) =
∣∣∣∣∣∣e3t e2tcos(t) e2tsin(t)0 −e2tsin(t) e2tcos(t)0 −e2tcos(t) −e2tsin(t)
∣∣∣∣∣∣ =
e3te2te2t
∣∣∣∣∣∣1 cos(t) sin(t)0 −sin(t) cos(t)0 −cos(t) −sin(t)
∣∣∣∣∣∣ =
e3te2te2t(sin2(t) + cos2(t)
)= e7t 6= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Hence the solutions x(1), x(2) and x(3) form a fundamental set, andthe general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Hence the solutions x(1), x(2) and x(3) form a fundamental set, andthe general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Hence the solutions x(1), x(2) and x(3) form a fundamental set, andthe general solution of the system is
X =
c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Hence the solutions x(1), x(2) and x(3) form a fundamental set, andthe general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Hence the solutions x(1), x(2) and x(3) form a fundamental set, andthe general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X =
c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Hence the solutions x(1), x(2) and x(3) form a fundamental set, andthe general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t +
c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Hence the solutions x(1), x(2) and x(3) form a fundamental set, andthe general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Hence the solutions x(1), x(2) and x(3) form a fundamental set, andthe general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Hence the solutions x(1), x(2) and x(3) form a fundamental set, andthe general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Hence the solutions x(1), x(2) and x(3) form a fundamental set, andthe general solution of the system is
X = c1x(1) + c2x(2) + c3x(3) =⇒
X = c1
100
e3t + c2
10−1
e2tcos(t)−
010
e2tsin(t)
+
c3
10−1
e2tcos(t) +
010
e2tsin(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
X =
x1x2x3
=
c1e3t + e2t(c2cos(t) + c3sin(t))
0 e2t(−c2sin(t) + c3cos(t))0 −e2t(c2cos(t) + c3sin(t))
Here is the direction field associated with the system
x ′1x ′2x ′3
=
3 1 10 2 10 −1 2
x1x2x3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
X =
x1x2x3
=
c1e3t + e2t(c2cos(t) + c3sin(t))
0 e2t(−c2sin(t) + c3cos(t))0 −e2t(c2cos(t) + c3sin(t))
Here is the direction field associated with the system
x ′1x ′2x ′3
=
3 1 10 2 10 −1 2
x1x2x3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
X =
x1x2x3
=
c1e3t + e2t(c2cos(t) + c3sin(t))
0 e2t(−c2sin(t) + c3cos(t))0 −e2t(c2cos(t) + c3sin(t))
Here is the direction field associated with the system
x ′1x ′2x ′3
=
3 1 10 2 10 −1 2
x1x2x3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
X =
x1x2x3
=
c1e3t + e2t(c2cos(t) + c3sin(t))
0 e2t(−c2sin(t) + c3cos(t))0 −e2t(c2cos(t) + c3sin(t))
Here is the direction field associated with the system
x ′1x ′2x ′3
=
3 1 10 2 10 −1 2
x1x2x3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
X =
x1x2x3
=
c1e3t + e2t(c2cos(t) + c3sin(t))
0 e2t(−c2sin(t) + c3cos(t))0 −e2t(c2cos(t) + c3sin(t))
Here is the direction field associated with the system
x ′1x ′2x ′3
=
3 1 10 2 10 −1 2
x1x2x3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
X =
x1x2x3
=
c1e3t + e2t(c2cos(t) + c3sin(t))
0 e2t(−c2sin(t) + c3cos(t))0 −e2t(c2cos(t) + c3sin(t))
Here is the direction field associated with the system
x ′1x ′2x ′3
=
3 1 10 2 10 −1 2
x1x2x3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
X =
x1x2x3
=
c1e3t + e2t(c2cos(t) + c3sin(t))
0 e2t(−c2sin(t) + c3cos(t))0 −e2t(c2cos(t) + c3sin(t))
Here is the direction field associated with the system
x ′1x ′2x ′3
=
3 1 10 2 10 −1 2
x1x2x3
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Example 7.18
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Example 7.18
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Example 7.18
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Example 7.18
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Example 7.18
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Example 7.18
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Example 7.18
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Example 7.18
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Example 7.18
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Example 7.18
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 =
(λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Example 7.18
Solve the following ODE
X′ = AX =
(−1/2 1−1 −1/2
)X
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣−1/2− λ 1−1 −1/2− λ
∣∣∣∣ = 0
(−1/2− λ)2 + 1 = (λ)2 + λ+5
4= 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
λ1 = −1
2+ i , λ2 = −1
2− i
If λ1 = −12 + i , then
(A− λ1I) x =
(−1/2− λ 1−1 −1/2− λ
)(v1v2
)=
(−1/2− (−1
2 + i) 1−1 −1/2− (−1
2 + i)
)(v1v2
)=(
−i 1−1 −i
)(v1v2
)=
(00
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
λ1 = −1
2+ i ,
λ2 = −1
2− i
If λ1 = −12 + i , then
(A− λ1I) x =
(−1/2− λ 1−1 −1/2− λ
)(v1v2
)=
(−1/2− (−1
2 + i) 1−1 −1/2− (−1
2 + i)
)(v1v2
)=(
−i 1−1 −i
)(v1v2
)=
(00
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
λ1 = −1
2+ i , λ2 = −1
2− i
If λ1 = −12 + i , then
(A− λ1I) x =
(−1/2− λ 1−1 −1/2− λ
)(v1v2
)=
(−1/2− (−1
2 + i) 1−1 −1/2− (−1
2 + i)
)(v1v2
)=(
−i 1−1 −i
)(v1v2
)=
(00
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
λ1 = −1
2+ i , λ2 = −1
2− i
If λ1 = −12 + i , then
(A− λ1I) x =
(−1/2− λ 1−1 −1/2− λ
)(v1v2
)=
(−1/2− (−1
2 + i) 1−1 −1/2− (−1
2 + i)
)(v1v2
)=(
−i 1−1 −i
)(v1v2
)=
(00
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
λ1 = −1
2+ i , λ2 = −1
2− i
If λ1 = −12 + i , then
(A− λ1I) x =
(−1/2− λ 1−1 −1/2− λ
)(v1v2
)=
(−1/2− (−1
2 + i) 1−1 −1/2− (−1
2 + i)
)(v1v2
)=(
−i 1−1 −i
)(v1v2
)=
(00
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
λ1 = −1
2+ i , λ2 = −1
2− i
If λ1 = −12 + i , then
(A− λ1I) x =
(−1/2− λ 1−1 −1/2− λ
)(v1v2
)=
(−1/2− (−1
2 + i) 1−1 −1/2− (−1
2 + i)
)(v1v2
)=(
−i 1−1 −i
)(v1v2
)=
(00
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
λ1 = −1
2+ i , λ2 = −1
2− i
If λ1 = −12 + i , then
(A− λ1I) x =
(−1/2− λ 1−1 −1/2− λ
)(v1v2
)=
(−1/2− (−1
2 + i) 1−1 −1/2− (−1
2 + i)
)(v1v2
)=
(−i 1−1 −i
)(v1v2
)=
(00
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
λ1 = −1
2+ i , λ2 = −1
2− i
If λ1 = −12 + i , then
(A− λ1I) x =
(−1/2− λ 1−1 −1/2− λ
)(v1v2
)=
(−1/2− (−1
2 + i) 1−1 −1/2− (−1
2 + i)
)(v1v2
)=(
−i 1−1 −i
)(v1v2
)=
(00
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
(1i
)If λ2 = −1
2 − i , then
(A− λ1I) x =
(−1/2− (−1
2 − i) 1−1 −1/2− (−1
2 − i)
)(v1v2
)=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
(1i
)If λ2 = −1
2 − i , then
(A− λ1I) x =
(−1/2− (−1
2 − i) 1−1 −1/2− (−1
2 − i)
)(v1v2
)=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
(1i
)
If λ2 = −12 − i , then
(A− λ1I) x =
(−1/2− (−1
2 − i) 1−1 −1/2− (−1
2 − i)
)(v1v2
)=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
(1i
)If λ2 = −1
2 − i , then
(A− λ1I) x =
(−1/2− (−1
2 − i) 1−1 −1/2− (−1
2 − i)
)(v1v2
)=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
(1i
)If λ2 = −1
2 − i , then
(A− λ1I) x =
(−1/2− (−1
2 − i) 1−1 −1/2− (−1
2 − i)
)(v1v2
)=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and a corresponding eigenvector is
v(1) =
(1i
)If λ2 = −1
2 − i , then
(A− λ1I) x =
(−1/2− (−1
2 − i) 1−1 −1/2− (−1
2 − i)
)(v1v2
)=
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding eigenvector is
v(2) =
(1−i
)The corresponding solutions of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ; x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding eigenvector is
v(2) =
(1−i
)The corresponding solutions of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ; x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding eigenvector is
v(2) =
(1−i
)The corresponding solutions of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ; x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding eigenvector is
v(2) =
(1−i
)
The corresponding solutions of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ; x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding eigenvector is
v(2) =
(1−i
)The corresponding solutions of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ; x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding eigenvector is
v(2) =
(1−i
)The corresponding solutions of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ;
x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding eigenvector is
v(2) =
(1−i
)The corresponding solutions of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ; x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
(i 1−1 i
)(v1v2
)=
and a corresponding eigenvector is
v(2) =
(1−i
)The corresponding solutions of the differential equation are
x(1) =
(1i
)e(−1/2+i)t ; x(2) =
(1
− i
)e(−1/2−i)t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence a set of real-valued solutions of is
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
To obtain a set of real-valued solutions,
we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence a set of real-valued solutions of is
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts
of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence a set of real-valued solutions of is
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2).
In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence a set of real-valued solutions of is
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence a set of real-valued solutions of is
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence a set of real-valued solutions of is
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence a set of real-valued solutions of is
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+
i
(e−t/2sin(t)
e−t/2cos(t)
)Hence a set of real-valued solutions of is
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)
Hence a set of real-valued solutions of is
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence a set of real-valued solutions of is
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence a set of real-valued solutions of is
u(t) = e−t/2(
cos(t)−sin(t)
)
v(t) = e−t/2(sin(t)cos(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
To obtain a set of real-valued solutions, we can choose the realand imaginary parts of either x (1) or x (2). In fact,
x(1) =
(1i
)e(−1/2+i)t =
(1i
)e−t/2 (cos(t) + i sin(t)) =
(e−t/2cos(t)
−e−t/2sin(t)
)+ i
(e−t/2sin(t)
e−t/2cos(t)
)Hence a set of real-valued solutions of is
u(t) = e−t/2(
cos(t)−sin(t)
)v(t) = e−t/2
(sin(t)cos(t)
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence the solutions x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence the solutions x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence the solutions x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence the solutions x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t
6= 0
Hence the solutions x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence the solutions x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
The Wronskian of these solutions is
W [x(1), x(2)](t) =
∣∣∣∣ e−t/2cos(t) e−t/2sin(t)
−e−t/2sin(t) e−t/2cos(t)
∣∣∣∣ =
e−t/2e−t/2∣∣∣∣ cos(t) sin(t)−sin(t) cos(t)
∣∣∣∣ = e−t 6= 0
Hence the solutions x(1), x(2) form a fundamental set,
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field associated with the system(
x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field associated with the system(
x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and the general solution of the system is
X =
c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field associated with the system(
x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) =
c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field associated with the system(
x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+
c2e−t/2
(sin(t)cos(t)
)Here is the direction field associated with the system(
x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)
Here is the direction field associated with the system(x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field associated with the system
(x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field associated with the system(
x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
and the general solution of the system is
X = c1x(1) + c2x(2) = c1e−t/2
(cos(t)−sin(t)
)+ c2e
−t/2(sin(t)cos(t)
)Here is the direction field associated with the system(
x ′1x ′2
)=
(−1/2 1−1 −1/2
)(x1x2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Complex Eigenvalues
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t)
form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions
on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β.
Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors
x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t),
is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system.
Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent
the matrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s start with the system
x′ = P(t)x
Suppose that x(1)(t), ..., x(n)(t) form a fundamental set ofsolutions on some interval α < t < β. Then the matrix
Ψ(t) =
x(1)1 · · · x
(n)1
......
x(1)n · · · x
(n)n
whose columns are the vectors x(1)(t), ..., x(n)(t), is said to be afundamental matrix for the linear system. Since the set ofsolutions is linearly independent the matrix is nonsingular.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
For instance, the system
x′ =
(1 14 1
)x
has solutions
x(1)(t) =
(e3t
2e3t
); x(2)(t) =
(e−t
−2e−t
)which are linearly independent. Thus a fundamental matrix for thesystem is
Ψ(t) =
(e3t e−t
2e3t −2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
For instance, the system
x′ =
(1 14 1
)x
has solutions
x(1)(t) =
(e3t
2e3t
); x(2)(t) =
(e−t
−2e−t
)which are linearly independent. Thus a fundamental matrix for thesystem is
Ψ(t) =
(e3t e−t
2e3t −2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
For instance, the system
x′ =
(1 14 1
)x
has solutions
x(1)(t) =
(e3t
2e3t
); x(2)(t) =
(e−t
−2e−t
)which are linearly independent. Thus a fundamental matrix for thesystem is
Ψ(t) =
(e3t e−t
2e3t −2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
For instance, the system
x′ =
(1 14 1
)x
has solutions
x(1)(t) =
(e3t
2e3t
); x(2)(t) =
(e−t
−2e−t
)which are linearly independent. Thus a fundamental matrix for thesystem is
Ψ(t) =
(e3t e−t
2e3t −2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
For instance, the system
x′ =
(1 14 1
)x
has solutions
x(1)(t) =
(e3t
2e3t
); x(2)(t) =
(e−t
−2e−t
)which are linearly independent. Thus a fundamental matrix for thesystem is
Ψ(t) =
(e3t e−t
2e3t −2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
For instance, the system
x′ =
(1 14 1
)x
has solutions
x(1)(t) =
(e3t
2e3t
); x(2)(t) =
(e−t
−2e−t
)which are linearly independent. Thus a fundamental matrix for thesystem is
Ψ(t) =
(e3t e−t
2e3t −2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
For instance, the system
x′ =
(1 14 1
)x
has solutions
x(1)(t) =
(e3t
2e3t
);
x(2)(t) =
(e−t
−2e−t
)which are linearly independent. Thus a fundamental matrix for thesystem is
Ψ(t) =
(e3t e−t
2e3t −2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
For instance, the system
x′ =
(1 14 1
)x
has solutions
x(1)(t) =
(e3t
2e3t
); x(2)(t) =
(e−t
−2e−t
)which are linearly independent. Thus a fundamental matrix for thesystem is
Ψ(t) =
(e3t e−t
2e3t −2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
For instance, the system
x′ =
(1 14 1
)x
has solutions
x(1)(t) =
(e3t
2e3t
); x(2)(t) =
(e−t
−2e−t
)
which are linearly independent. Thus a fundamental matrix for thesystem is
Ψ(t) =
(e3t e−t
2e3t −2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
For instance, the system
x′ =
(1 14 1
)x
has solutions
x(1)(t) =
(e3t
2e3t
); x(2)(t) =
(e−t
−2e−t
)which are linearly independent.
Thus a fundamental matrix for thesystem is
Ψ(t) =
(e3t e−t
2e3t −2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
For instance, the system
x′ =
(1 14 1
)x
has solutions
x(1)(t) =
(e3t
2e3t
); x(2)(t) =
(e−t
−2e−t
)which are linearly independent. Thus a fundamental matrix for thesystem is
Ψ(t) =
(e3t e−t
2e3t −2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
For instance, the system
x′ =
(1 14 1
)x
has solutions
x(1)(t) =
(e3t
2e3t
); x(2)(t) =
(e−t
−2e−t
)which are linearly independent. Thus a fundamental matrix for thesystem is
Ψ(t) =
(e3t e−t
2e3t −2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
For instance, the system
x′ =
(1 14 1
)x
has solutions
x(1)(t) =
(e3t
2e3t
); x(2)(t) =
(e−t
−2e−t
)which are linearly independent. Thus a fundamental matrix for thesystem is
Ψ(t) =
(e3t e−t
2e3t −2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Using a fundamental matrix the general solution can be written as
x = Ψ(t)c; c = constant
and if we imposed initial conditions x(t0) = x0, where t0 is a givenpoint on α < t < β and x0 is given initial vector, we obtain
Ψ(t0)c = x0
c = Ψ−1(t0)x0
x = Ψ(t)Ψ−1(t0)x0
is the solution of the initial value problem
x′ = P(t)x; x(t0) = x0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Using a fundamental matrix the general solution can be written as
x = Ψ(t)c; c = constant
and if we imposed initial conditions x(t0) = x0, where t0 is a givenpoint on α < t < β and x0 is given initial vector, we obtain
Ψ(t0)c = x0
c = Ψ−1(t0)x0
x = Ψ(t)Ψ−1(t0)x0
is the solution of the initial value problem
x′ = P(t)x; x(t0) = x0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Using a fundamental matrix the general solution can be written as
x = Ψ(t)c; c = constant
and if we imposed initial conditions x(t0) = x0, where t0 is a givenpoint on α < t < β and x0 is given initial vector, we obtain
Ψ(t0)c = x0
c = Ψ−1(t0)x0
x = Ψ(t)Ψ−1(t0)x0
is the solution of the initial value problem
x′ = P(t)x; x(t0) = x0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Using a fundamental matrix the general solution can be written as
x = Ψ(t)c; c = constant
and if we imposed initial conditions
x(t0) = x0, where t0 is a givenpoint on α < t < β and x0 is given initial vector, we obtain
Ψ(t0)c = x0
c = Ψ−1(t0)x0
x = Ψ(t)Ψ−1(t0)x0
is the solution of the initial value problem
x′ = P(t)x; x(t0) = x0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Using a fundamental matrix the general solution can be written as
x = Ψ(t)c; c = constant
and if we imposed initial conditions x(t0) = x0,
where t0 is a givenpoint on α < t < β and x0 is given initial vector, we obtain
Ψ(t0)c = x0
c = Ψ−1(t0)x0
x = Ψ(t)Ψ−1(t0)x0
is the solution of the initial value problem
x′ = P(t)x; x(t0) = x0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Using a fundamental matrix the general solution can be written as
x = Ψ(t)c; c = constant
and if we imposed initial conditions x(t0) = x0, where t0 is a givenpoint on α < t < β and
x0 is given initial vector, we obtain
Ψ(t0)c = x0
c = Ψ−1(t0)x0
x = Ψ(t)Ψ−1(t0)x0
is the solution of the initial value problem
x′ = P(t)x; x(t0) = x0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Using a fundamental matrix the general solution can be written as
x = Ψ(t)c; c = constant
and if we imposed initial conditions x(t0) = x0, where t0 is a givenpoint on α < t < β and x0 is given initial vector,
we obtain
Ψ(t0)c = x0
c = Ψ−1(t0)x0
x = Ψ(t)Ψ−1(t0)x0
is the solution of the initial value problem
x′ = P(t)x; x(t0) = x0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Using a fundamental matrix the general solution can be written as
x = Ψ(t)c; c = constant
and if we imposed initial conditions x(t0) = x0, where t0 is a givenpoint on α < t < β and x0 is given initial vector, we obtain
Ψ(t0)c = x0
c = Ψ−1(t0)x0
x = Ψ(t)Ψ−1(t0)x0
is the solution of the initial value problem
x′ = P(t)x; x(t0) = x0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Using a fundamental matrix the general solution can be written as
x = Ψ(t)c; c = constant
and if we imposed initial conditions x(t0) = x0, where t0 is a givenpoint on α < t < β and x0 is given initial vector, we obtain
Ψ(t0)c = x0
c = Ψ−1(t0)x0
x = Ψ(t)Ψ−1(t0)x0
is the solution of the initial value problem
x′ = P(t)x; x(t0) = x0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Using a fundamental matrix the general solution can be written as
x = Ψ(t)c; c = constant
and if we imposed initial conditions x(t0) = x0, where t0 is a givenpoint on α < t < β and x0 is given initial vector, we obtain
Ψ(t0)c = x0
c = Ψ−1(t0)x0
x = Ψ(t)Ψ−1(t0)x0
is the solution of the initial value problem
x′ = P(t)x; x(t0) = x0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Using a fundamental matrix the general solution can be written as
x = Ψ(t)c; c = constant
and if we imposed initial conditions x(t0) = x0, where t0 is a givenpoint on α < t < β and x0 is given initial vector, we obtain
Ψ(t0)c = x0
c = Ψ−1(t0)x0
x = Ψ(t)Ψ−1(t0)x0
is the solution of the initial value problem
x′ = P(t)x; x(t0) = x0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Using a fundamental matrix the general solution can be written as
x = Ψ(t)c; c = constant
and if we imposed initial conditions x(t0) = x0, where t0 is a givenpoint on α < t < β and x0 is given initial vector, we obtain
Ψ(t0)c = x0
c = Ψ−1(t0)x0
x = Ψ(t)Ψ−1(t0)x0
is the solution of the initial value problem
x′ = P(t)x; x(t0) = x0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Using a fundamental matrix the general solution can be written as
x = Ψ(t)c; c = constant
and if we imposed initial conditions x(t0) = x0, where t0 is a givenpoint on α < t < β and x0 is given initial vector, we obtain
Ψ(t0)c = x0
c = Ψ−1(t0)x0
x = Ψ(t)Ψ−1(t0)x0
is the solution of the initial value problem
x′ = P(t)x;
x(t0) = x0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Using a fundamental matrix the general solution can be written as
x = Ψ(t)c; c = constant
and if we imposed initial conditions x(t0) = x0, where t0 is a givenpoint on α < t < β and x0 is given initial vector, we obtain
Ψ(t0)c = x0
c = Ψ−1(t0)x0
x = Ψ(t)Ψ−1(t0)x0
is the solution of the initial value problem
x′ = P(t)x; x(t0) = x0Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
Sometimes it is convenient to make use of the specialfundamental matrix , denoted by Φ, such that the initialcondition
x(j) = e(j); e(j) =
0...1...0
j − th row
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Recall that each column
of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
Sometimes it is convenient to make use of the specialfundamental matrix , denoted by Φ, such that the initialcondition
x(j) = e(j); e(j) =
0...1...0
j − th row
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Recall that each column of the fundamental matrix Ψ(t)
is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
Sometimes it is convenient to make use of the specialfundamental matrix , denoted by Φ, such that the initialcondition
x(j) = e(j); e(j) =
0...1...0
j − th row
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE.
It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
Sometimes it is convenient to make use of the specialfundamental matrix , denoted by Φ, such that the initialcondition
x(j) = e(j); e(j) =
0...1...0
j − th row
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t)
satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
Sometimes it is convenient to make use of the specialfundamental matrix , denoted by Φ, such that the initialcondition
x(j) = e(j); e(j) =
0...1...0
j − th row
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
Sometimes it is convenient to make use of the specialfundamental matrix , denoted by Φ, such that the initialcondition
x(j) = e(j); e(j) =
0...1...0
j − th row
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
Sometimes it is convenient to make use of the specialfundamental matrix , denoted by Φ, such that the initialcondition
x(j) = e(j); e(j) =
0...1...0
j − th row
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
Sometimes it is convenient
to make use of the specialfundamental matrix , denoted by Φ, such that the initialcondition
x(j) = e(j); e(j) =
0...1...0
j − th row
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
Sometimes it is convenient to make use of the specialfundamental matrix ,
denoted by Φ, such that the initialcondition
x(j) = e(j); e(j) =
0...1...0
j − th row
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
Sometimes it is convenient to make use of the specialfundamental matrix , denoted by Φ, such that the initialcondition
x(j) = e(j); e(j) =
0...1...0
j − th row
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
Sometimes it is convenient to make use of the specialfundamental matrix , denoted by Φ, such that the initialcondition
x(j) = e(j);
e(j) =
0...1...0
j − th row
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
Sometimes it is convenient to make use of the specialfundamental matrix , denoted by Φ, such that the initialcondition
x(j) = e(j); e(j) =
0...1...0
j − th row
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Recall that each column of the fundamental matrix Ψ(t) is asolution of the ODE. It follows that Ψ(t) satisfies the matrixdifferential equation
Ψ′ = P(t)Ψ
Sometimes it is convenient to make use of the specialfundamental matrix , denoted by Φ, such that the initialcondition
x(j) = e(j); e(j) =
0...1...0
j − th row
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus Φ(t) has the property that
Φ(t0) =
1 0 · · · 00 1 · · · 0...
......
0 0 · · · 1
= I
and the solution of the IVP is given by
x = Φ(t)Φ−1(t0)x0 = Φ(t)x0
in another words
Φ(t) = Ψ(t)Ψ−1(t0)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus Φ(t) has the property that
Φ(t0) =
1 0 · · · 00 1 · · · 0...
......
0 0 · · · 1
= I
and the solution of the IVP is given by
x = Φ(t)Φ−1(t0)x0 = Φ(t)x0
in another words
Φ(t) = Ψ(t)Ψ−1(t0)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus Φ(t) has the property that
Φ(t0) =
1 0 · · · 00 1 · · · 0...
......
0 0 · · · 1
= I
and the solution of the IVP is given by
x = Φ(t)Φ−1(t0)x0 = Φ(t)x0
in another words
Φ(t) = Ψ(t)Ψ−1(t0)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus Φ(t) has the property that
Φ(t0) =
1 0 · · · 00 1 · · · 0...
......
0 0 · · · 1
=
I
and the solution of the IVP is given by
x = Φ(t)Φ−1(t0)x0 = Φ(t)x0
in another words
Φ(t) = Ψ(t)Ψ−1(t0)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus Φ(t) has the property that
Φ(t0) =
1 0 · · · 00 1 · · · 0...
......
0 0 · · · 1
= I
and the solution of the IVP is given by
x = Φ(t)Φ−1(t0)x0 = Φ(t)x0
in another words
Φ(t) = Ψ(t)Ψ−1(t0)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus Φ(t) has the property that
Φ(t0) =
1 0 · · · 00 1 · · · 0...
......
0 0 · · · 1
= I
and the solution of the IVP is given by
x = Φ(t)Φ−1(t0)x0 = Φ(t)x0
in another words
Φ(t) = Ψ(t)Ψ−1(t0)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus Φ(t) has the property that
Φ(t0) =
1 0 · · · 00 1 · · · 0...
......
0 0 · · · 1
= I
and the solution of the IVP is given by
x =
Φ(t)Φ−1(t0)x0 = Φ(t)x0
in another words
Φ(t) = Ψ(t)Ψ−1(t0)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus Φ(t) has the property that
Φ(t0) =
1 0 · · · 00 1 · · · 0...
......
0 0 · · · 1
= I
and the solution of the IVP is given by
x = Φ(t)Φ−1(t0)x0 =
Φ(t)x0
in another words
Φ(t) = Ψ(t)Ψ−1(t0)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus Φ(t) has the property that
Φ(t0) =
1 0 · · · 00 1 · · · 0...
......
0 0 · · · 1
= I
and the solution of the IVP is given by
x = Φ(t)Φ−1(t0)x0 = Φ(t)x0
in another words
Φ(t) = Ψ(t)Ψ−1(t0)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus Φ(t) has the property that
Φ(t0) =
1 0 · · · 00 1 · · · 0...
......
0 0 · · · 1
= I
and the solution of the IVP is given by
x = Φ(t)Φ−1(t0)x0 = Φ(t)x0
in another words
Φ(t) = Ψ(t)Ψ−1(t0)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus Φ(t) has the property that
Φ(t0) =
1 0 · · · 00 1 · · · 0...
......
0 0 · · · 1
= I
and the solution of the IVP is given by
x = Φ(t)Φ−1(t0)x0 = Φ(t)x0
in another words
Φ(t) =
Ψ(t)Ψ−1(t0)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus Φ(t) has the property that
Φ(t0) =
1 0 · · · 00 1 · · · 0...
......
0 0 · · · 1
= I
and the solution of the IVP is given by
x = Φ(t)Φ−1(t0)x0 = Φ(t)x0
in another words
Φ(t) = Ψ(t)Ψ−1(t0)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus, for instance the system
x′ =
(1 14 1
)x
subject to the different initial conditions
x(1)(0) =
(10
); x(2)(0) =
(01
)has the particular solutions equal to
x(t) =1
2
(e3t
2e3t
)+
1
2
(e−t
−2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus, for instance the system
x′ =
(1 14 1
)x
subject to the different initial conditions
x(1)(0) =
(10
); x(2)(0) =
(01
)has the particular solutions equal to
x(t) =1
2
(e3t
2e3t
)+
1
2
(e−t
−2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus, for instance the system
x′ =
(1 14 1
)x
subject to the different initial conditions
x(1)(0) =
(10
); x(2)(0) =
(01
)has the particular solutions equal to
x(t) =1
2
(e3t
2e3t
)+
1
2
(e−t
−2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus, for instance the system
x′ =
(1 14 1
)x
subject to the different initial conditions
x(1)(0) =
(10
); x(2)(0) =
(01
)has the particular solutions equal to
x(t) =1
2
(e3t
2e3t
)+
1
2
(e−t
−2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus, for instance the system
x′ =
(1 14 1
)x
subject to the different initial conditions
x(1)(0) =
(10
); x(2)(0) =
(01
)has the particular solutions equal to
x(t) =1
2
(e3t
2e3t
)+
1
2
(e−t
−2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus, for instance the system
x′ =
(1 14 1
)x
subject to the different initial conditions
x(1)(0) =
(10
); x(2)(0) =
(01
)has the particular solutions equal to
x(t) =1
2
(e3t
2e3t
)+
1
2
(e−t
−2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus, for instance the system
x′ =
(1 14 1
)x
subject to the different initial conditions
x(1)(0) =
(10
);
x(2)(0) =
(01
)has the particular solutions equal to
x(t) =1
2
(e3t
2e3t
)+
1
2
(e−t
−2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus, for instance the system
x′ =
(1 14 1
)x
subject to the different initial conditions
x(1)(0) =
(10
); x(2)(0) =
(01
)has the particular solutions equal to
x(t) =1
2
(e3t
2e3t
)+
1
2
(e−t
−2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus, for instance the system
x′ =
(1 14 1
)x
subject to the different initial conditions
x(1)(0) =
(10
); x(2)(0) =
(01
)
has the particular solutions equal to
x(t) =1
2
(e3t
2e3t
)+
1
2
(e−t
−2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus, for instance the system
x′ =
(1 14 1
)x
subject to the different initial conditions
x(1)(0) =
(10
); x(2)(0) =
(01
)has the particular solutions equal to
x(t) =1
2
(e3t
2e3t
)+
1
2
(e−t
−2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus, for instance the system
x′ =
(1 14 1
)x
subject to the different initial conditions
x(1)(0) =
(10
); x(2)(0) =
(01
)has the particular solutions equal to
x(t) =
1
2
(e3t
2e3t
)+
1
2
(e−t
−2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus, for instance the system
x′ =
(1 14 1
)x
subject to the different initial conditions
x(1)(0) =
(10
); x(2)(0) =
(01
)has the particular solutions equal to
x(t) =1
2
(e3t
2e3t
)+
1
2
(e−t
−2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Thus, for instance the system
x′ =
(1 14 1
)x
subject to the different initial conditions
x(1)(0) =
(10
); x(2)(0) =
(01
)has the particular solutions equal to
x(t) =1
2
(e3t
2e3t
)+
1
2
(e−t
−2e−t
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
x(t) =1
4
(e3t
2e3t
)− 1
4
(e−t
−2e−t
)Hence
Φ(t) =
12e
3t + 12e−t 1
2e3t − 1
2e−t
e3t − e−t 12e
3t + 12e−t
OBS
Note that the elements of Φ(t) are more complicated than those ofthe fundamental matrix Ψ(t); however, it is now easy to determinethe solution corresponding to any set of initial conditions.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
x(t) =
1
4
(e3t
2e3t
)− 1
4
(e−t
−2e−t
)Hence
Φ(t) =
12e
3t + 12e−t 1
2e3t − 1
2e−t
e3t − e−t 12e
3t + 12e−t
OBS
Note that the elements of Φ(t) are more complicated than those ofthe fundamental matrix Ψ(t); however, it is now easy to determinethe solution corresponding to any set of initial conditions.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
x(t) =1
4
(e3t
2e3t
)−
1
4
(e−t
−2e−t
)Hence
Φ(t) =
12e
3t + 12e−t 1
2e3t − 1
2e−t
e3t − e−t 12e
3t + 12e−t
OBS
Note that the elements of Φ(t) are more complicated than those ofthe fundamental matrix Ψ(t); however, it is now easy to determinethe solution corresponding to any set of initial conditions.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
x(t) =1
4
(e3t
2e3t
)− 1
4
(e−t
−2e−t
)
Hence
Φ(t) =
12e
3t + 12e−t 1
2e3t − 1
2e−t
e3t − e−t 12e
3t + 12e−t
OBS
Note that the elements of Φ(t) are more complicated than those ofthe fundamental matrix Ψ(t); however, it is now easy to determinethe solution corresponding to any set of initial conditions.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
x(t) =1
4
(e3t
2e3t
)− 1
4
(e−t
−2e−t
)Hence
Φ(t) =
12e
3t + 12e−t 1
2e3t − 1
2e−t
e3t − e−t 12e
3t + 12e−t
OBS
Note that the elements of Φ(t) are more complicated than those ofthe fundamental matrix Ψ(t); however, it is now easy to determinethe solution corresponding to any set of initial conditions.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
x(t) =1
4
(e3t
2e3t
)− 1
4
(e−t
−2e−t
)Hence
Φ(t) =
12e
3t + 12e−t 1
2e3t − 1
2e−t
e3t − e−t 12e
3t + 12e−t
OBS
Note that the elements of Φ(t) are more complicated than those ofthe fundamental matrix Ψ(t); however, it is now easy to determinethe solution corresponding to any set of initial conditions.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
x(t) =1
4
(e3t
2e3t
)− 1
4
(e−t
−2e−t
)Hence
Φ(t) =
12e
3t + 12e−t 1
2e3t − 1
2e−t
e3t − e−t 12e
3t + 12e−t
OBS
Note that the elements of Φ(t) are more complicated than those ofthe fundamental matrix Ψ(t); however, it is now easy to determinethe solution corresponding to any set of initial conditions.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
x(t) =1
4
(e3t
2e3t
)− 1
4
(e−t
−2e−t
)Hence
Φ(t) =
12e
3t + 12e−t 1
2e3t − 1
2e−t
e3t − e−t 12e
3t + 12e−t
OBS
Note that the elements of Φ(t) are more complicated than those ofthe fundamental matrix Ψ(t); however, it is now easy to determinethe solution corresponding to any set of initial conditions.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
x(t) =1
4
(e3t
2e3t
)− 1
4
(e−t
−2e−t
)Hence
Φ(t) =
12e
3t + 12e−t 1
2e3t − 1
2e−t
e3t − e−t 12e
3t + 12e−t
OBS
Note that the elements of Φ(t)
are more complicated than those ofthe fundamental matrix Ψ(t); however, it is now easy to determinethe solution corresponding to any set of initial conditions.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
x(t) =1
4
(e3t
2e3t
)− 1
4
(e−t
−2e−t
)Hence
Φ(t) =
12e
3t + 12e−t 1
2e3t − 1
2e−t
e3t − e−t 12e
3t + 12e−t
OBS
Note that the elements of Φ(t) are more complicated
than those ofthe fundamental matrix Ψ(t); however, it is now easy to determinethe solution corresponding to any set of initial conditions.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
x(t) =1
4
(e3t
2e3t
)− 1
4
(e−t
−2e−t
)Hence
Φ(t) =
12e
3t + 12e−t 1
2e3t − 1
2e−t
e3t − e−t 12e
3t + 12e−t
OBS
Note that the elements of Φ(t) are more complicated than those ofthe fundamental matrix Ψ(t);
however, it is now easy to determinethe solution corresponding to any set of initial conditions.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
x(t) =1
4
(e3t
2e3t
)− 1
4
(e−t
−2e−t
)Hence
Φ(t) =
12e
3t + 12e−t 1
2e3t − 1
2e−t
e3t − e−t 12e
3t + 12e−t
OBS
Note that the elements of Φ(t) are more complicated than those ofthe fundamental matrix Ψ(t); however, it is now easy to determinethe solution
corresponding to any set of initial conditions.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
x(t) =1
4
(e3t
2e3t
)− 1
4
(e−t
−2e−t
)Hence
Φ(t) =
12e
3t + 12e−t 1
2e3t − 1
2e−t
e3t − e−t 12e
3t + 12e−t
OBS
Note that the elements of Φ(t) are more complicated than those ofthe fundamental matrix Ψ(t); however, it is now easy to determinethe solution corresponding to any set of initial conditions.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem
for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
The Matrix eAt
Recall that the solution of the initial value problem
x ′ = ax , x(0) = x0, a = constant
is given by
x(t) = x0eat
Now, consider the corresponding initial value problem for an n × nsystem
x′ = Ax, x(0) = x0
where A is a constant matrix.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Applying the results of already obtained,
we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I.
Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat .
let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat
can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t.
Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and
consider the corresponding series
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Applying the results of already obtained, we can write its solutionas
x = Φ(t)x0
where Φ(0) = I. Thus, Φ(t), is playing the roll of eat . let’s seethis with more detail.
The scalar exponential function eat can be represented by thepower series
eat = 1 +∞∑n=1
antn
n!
which converges for all t. Let us now replace the scalar a by then × n constant matrix A and consider the corresponding series
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!=
I + At +A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I +
At +A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+
...+Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+
...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix.
It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat
each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges
for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞.
Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix,
which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series
term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term,
we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=
∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!=
A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]=
AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
I +∞∑n=1
Antn
n!= I + At +
A2t2
2!+ ...+
Ant2
n!+ ...
Each term in the series is an n × n matrix. It is possible to showthat each element of this matrix sum converges for all t asn→∞. Thus, we have a well defined n × n matrix, which will bedenote by eAt
eAt = I +∞∑n=1
Antn
n!
By differentiating the above series term by term, we obtain
d
dt
[eAt]
=∞∑n=1
Antn−1
(n − 1)!= A
[I +
∞∑n=1
Antn
n!
]= AeAt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the fundamental matrix Φ satisfies thesame initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the fundamental matrix Φ satisfies thesame initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the fundamental matrix Φ satisfies thesame initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0
in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the fundamental matrix Φ satisfies thesame initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt
we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the fundamental matrix Φ satisfies thesame initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the fundamental matrix Φ satisfies thesame initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the fundamental matrix Φ satisfies thesame initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way,
we have that the fundamental matrix Φ satisfies thesame initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that
the fundamental matrix Φ satisfies thesame initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the fundamental matrix Φ
satisfies thesame initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the fundamental matrix Φ satisfies thesame initial value problem as
eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the fundamental matrix Φ satisfies thesame initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Therefore, eAt satisfies the differential equation
d
dt
[eAt]
= AeAt
Further, by setting t = 0 in the definition of eAt we find that eAt
satisfies the initial condition
eAt∣∣∣t=0
= I
In this way, we have that the fundamental matrix Φ satisfies thesame initial value problem as eAt , namely,
Φ′ = AΦ, Φ(0) = I
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the fundamental matrix Φ(t)are the same. Thus we can write the solution of the initial valueproblem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Then, by uniqueness of an IVP
(extended to matrix differentialequations), we conclude that eAt and the fundamental matrix Φ(t)are the same. Thus we can write the solution of the initial valueproblem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Then, by uniqueness of an IVP (extended to matrix differentialequations),
we conclude that eAt and the fundamental matrix Φ(t)are the same. Thus we can write the solution of the initial valueproblem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and
the fundamental matrix Φ(t)are the same. Thus we can write the solution of the initial valueproblem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the fundamental matrix Φ(t)are the same.
Thus we can write the solution of the initial valueproblem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the fundamental matrix Φ(t)are the same. Thus we can write the solution
of the initial valueproblem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the fundamental matrix Φ(t)are the same. Thus we can write the solution of the initial valueproblem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the fundamental matrix Φ(t)are the same. Thus we can write the solution of the initial valueproblem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the fundamental matrix Φ(t)are the same. Thus we can write the solution of the initial valueproblem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Then, by uniqueness of an IVP (extended to matrix differentialequations), we conclude that eAt and the fundamental matrix Φ(t)are the same. Thus we can write the solution of the initial valueproblem
x = Ax, x(0) = x0
in the form
x = eAtx0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty
is that the equations are usuallycoupled.
Hence the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence the equations in the system must be solved simultaneously.
On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled,
then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into
an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable )
corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix.
Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Diagonalizable Matrices.
The basic reason why a system of linear (algebraic or differential)equations presents some difficulty is that the equations are usuallycoupled.
Hence the equations in the system must be solved simultaneously.On the contrary, if the system is uncoupled, then each equationcan be solved independently of all the others.
Transforming the coupled system into an equivalent uncoupledsystem ( in which each equation contains only one unknownvariable ) corresponds to transforming the coefficient matrix A intoa diagonal matrix. Eigenvectors are useful in accomplishing such atransformation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s assume that the matrix A has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
T =
x(1)1 · · · x(n)
......
x(1)n · · · x
(n)n
we have
AT =
Ax(1) · · · Ax
(n)1
......
... · · ·...
=
λ1x
(1)1 · · · λnx
(n)1
λ1x(1)2
...
λ1x(1)n λnx
(n)n
= TD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s assume that the matrix A
has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
T =
x(1)1 · · · x(n)
......
x(1)n · · · x
(n)n
we have
AT =
Ax(1) · · · Ax
(n)1
......
... · · ·...
=
λ1x
(1)1 · · · λnx
(n)1
λ1x(1)2
...
λ1x(1)n λnx
(n)n
= TD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s assume that the matrix A has n eigenvectors x(1), x(2), ...,x(n)
linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
T =
x(1)1 · · · x(n)
......
x(1)n · · · x
(n)n
we have
AT =
Ax(1) · · · Ax
(n)1
......
... · · ·...
=
λ1x
(1)1 · · · λnx
(n)1
λ1x(1)2
...
λ1x(1)n λnx
(n)n
= TD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s assume that the matrix A has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
T =
x(1)1 · · · x(n)
......
x(1)n · · · x
(n)n
we have
AT =
Ax(1) · · · Ax
(n)1
......
... · · ·...
=
λ1x
(1)1 · · · λnx
(n)1
λ1x(1)2
...
λ1x(1)n λnx
(n)n
= TD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s assume that the matrix A has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
T =
x(1)1 · · · x(n)
......
x(1)n · · · x
(n)n
we have
AT =
Ax(1) · · · Ax
(n)1
......
... · · ·...
=
λ1x
(1)1 · · · λnx
(n)1
λ1x(1)2
...
λ1x(1)n λnx
(n)n
= TD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s assume that the matrix A has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
T =
x(1)1 · · · x(n)
......
x(1)n · · · x
(n)n
we have
AT =
Ax(1) · · · Ax
(n)1
......
... · · ·...
=
λ1x
(1)1 · · · λnx
(n)1
λ1x(1)2
...
λ1x(1)n λnx
(n)n
= TD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s assume that the matrix A has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
T =
x(1)1 · · · x(n)
......
x(1)n · · · x
(n)n
we have
AT =
Ax(1) · · · Ax
(n)1
......
... · · ·...
=
λ1x
(1)1 · · · λnx
(n)1
λ1x(1)2
...
λ1x(1)n λnx
(n)n
= TD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s assume that the matrix A has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
T =
x(1)1 · · · x(n)
......
x(1)n · · · x
(n)n
we have
AT =
Ax(1) · · · Ax
(n)1
......
... · · ·...
=
λ1x
(1)1 · · · λnx
(n)1
λ1x(1)2
...
λ1x(1)n λnx
(n)n
= TD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s assume that the matrix A has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
T =
x(1)1 · · · x(n)
......
x(1)n · · · x
(n)n
we have
AT =
Ax(1) · · · Ax
(n)1
......
... · · ·...
=
λ1x
(1)1 · · · λnx
(n)1
λ1x(1)2
...
λ1x(1)n λnx
(n)n
= TD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s assume that the matrix A has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
T =
x(1)1 · · · x(n)
......
x(1)n · · · x
(n)n
we have
AT =
Ax(1) · · · Ax
(n)1
......
... · · ·...
=
λ1x
(1)1 · · · λnx
(n)1
λ1x(1)2
...
λ1x(1)n λnx
(n)n
= TD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s assume that the matrix A has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
T =
x(1)1 · · · x(n)
......
x(1)n · · · x
(n)n
we have
AT =
Ax(1) · · · Ax
(n)1
......
... · · ·...
=
λ1x
(1)1 · · · λnx
(n)1
λ1x(1)2
...
λ1x(1)n λnx
(n)n
=
TD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
Let’s assume that the matrix A has n eigenvectors x(1), x(2), ...,x(n) linearly indepedent, then
Ax(1) = λ1x(1); Ax(2) = λ2x(2); ...Ax(n) = λnx(n)
and considering the matrix
T =
x(1)1 · · · x(n)
......
x(1)n · · · x
(n)n
we have
AT =
Ax(1) · · · Ax
(n)1
......
... · · ·...
=
λ1x
(1)1 · · · λnx
(n)1
λ1x(1)2
...
λ1x(1)n λnx
(n)n
= TD
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
T−1AT = D
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
T−1AT = D
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
T−1AT = D
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
T−1AT = D
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
T−1AT = D
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
T−1AT = D
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
T−1AT = D
Thus, if the eigenvalues and eigenvectors of A are known,
A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
T−1AT = D
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix
by the process shown in theabove equation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
where D is the diagonal matrix
D =
λ1
λ2. . .
λn
whose diagonal elements are the eigenvalues of A. From the lastequations we have that it follows that
T−1AT = D
Thus, if the eigenvalues and eigenvectors of A are known, A canbe transformed into a diagonal matrix by the process shown in theabove equation.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental Matrices
This process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian, then the determination of T−1 is very simple.The eigenvectors v(1), ..., v(n) of A are known to be mutuallyorthogonal, so let us choose them so that they are also normalizedby < v(i), v(i) >= 1 for each i . It is easy verify that T−1 = T∗. Inother words, the inverse of T is the same as its adjoint (thetranspose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix T such that T−1AT = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental MatricesThis process is known as a similarity transformation.
Alternatively, we may say that A is diagonalizable.
If A is Hermitian, then the determination of T−1 is very simple.The eigenvectors v(1), ..., v(n) of A are known to be mutuallyorthogonal, so let us choose them so that they are also normalizedby < v(i), v(i) >= 1 for each i . It is easy verify that T−1 = T∗. Inother words, the inverse of T is the same as its adjoint (thetranspose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix T such that T−1AT = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental MatricesThis process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian, then the determination of T−1 is very simple.The eigenvectors v(1), ..., v(n) of A are known to be mutuallyorthogonal, so let us choose them so that they are also normalizedby < v(i), v(i) >= 1 for each i . It is easy verify that T−1 = T∗. Inother words, the inverse of T is the same as its adjoint (thetranspose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix T such that T−1AT = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental MatricesThis process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian,
then the determination of T−1 is very simple.The eigenvectors v(1), ..., v(n) of A are known to be mutuallyorthogonal, so let us choose them so that they are also normalizedby < v(i), v(i) >= 1 for each i . It is easy verify that T−1 = T∗. Inother words, the inverse of T is the same as its adjoint (thetranspose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix T such that T−1AT = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental MatricesThis process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian, then the determination of T−1 is very simple.
The eigenvectors v(1), ..., v(n) of A are known to be mutuallyorthogonal, so let us choose them so that they are also normalizedby < v(i), v(i) >= 1 for each i . It is easy verify that T−1 = T∗. Inother words, the inverse of T is the same as its adjoint (thetranspose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix T such that T−1AT = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental MatricesThis process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian, then the determination of T−1 is very simple.The eigenvectors v(1), ..., v(n)
of A are known to be mutuallyorthogonal, so let us choose them so that they are also normalizedby < v(i), v(i) >= 1 for each i . It is easy verify that T−1 = T∗. Inother words, the inverse of T is the same as its adjoint (thetranspose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix T such that T−1AT = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental MatricesThis process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian, then the determination of T−1 is very simple.The eigenvectors v(1), ..., v(n) of A are known to be mutuallyorthogonal,
so let us choose them so that they are also normalizedby < v(i), v(i) >= 1 for each i . It is easy verify that T−1 = T∗. Inother words, the inverse of T is the same as its adjoint (thetranspose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix T such that T−1AT = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental MatricesThis process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian, then the determination of T−1 is very simple.The eigenvectors v(1), ..., v(n) of A are known to be mutuallyorthogonal, so let us choose them so that
they are also normalizedby < v(i), v(i) >= 1 for each i . It is easy verify that T−1 = T∗. Inother words, the inverse of T is the same as its adjoint (thetranspose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix T such that T−1AT = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental MatricesThis process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian, then the determination of T−1 is very simple.The eigenvectors v(1), ..., v(n) of A are known to be mutuallyorthogonal, so let us choose them so that they are also normalizedby < v(i), v(i) >= 1
for each i . It is easy verify that T−1 = T∗. Inother words, the inverse of T is the same as its adjoint (thetranspose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix T such that T−1AT = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental MatricesThis process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian, then the determination of T−1 is very simple.The eigenvectors v(1), ..., v(n) of A are known to be mutuallyorthogonal, so let us choose them so that they are also normalizedby < v(i), v(i) >= 1 for each i .
It is easy verify that T−1 = T∗. Inother words, the inverse of T is the same as its adjoint (thetranspose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix T such that T−1AT = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental MatricesThis process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian, then the determination of T−1 is very simple.The eigenvectors v(1), ..., v(n) of A are known to be mutuallyorthogonal, so let us choose them so that they are also normalizedby < v(i), v(i) >= 1 for each i . It is easy verify that T−1 = T∗.
Inother words, the inverse of T is the same as its adjoint (thetranspose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix T such that T−1AT = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental MatricesThis process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian, then the determination of T−1 is very simple.The eigenvectors v(1), ..., v(n) of A are known to be mutuallyorthogonal, so let us choose them so that they are also normalizedby < v(i), v(i) >= 1 for each i . It is easy verify that T−1 = T∗. Inother words,
the inverse of T is the same as its adjoint (thetranspose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix T such that T−1AT = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental MatricesThis process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian, then the determination of T−1 is very simple.The eigenvectors v(1), ..., v(n) of A are known to be mutuallyorthogonal, so let us choose them so that they are also normalizedby < v(i), v(i) >= 1 for each i . It is easy verify that T−1 = T∗. Inother words, the inverse of T
is the same as its adjoint (thetranspose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix T such that T−1AT = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental MatricesThis process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian, then the determination of T−1 is very simple.The eigenvectors v(1), ..., v(n) of A are known to be mutuallyorthogonal, so let us choose them so that they are also normalizedby < v(i), v(i) >= 1 for each i . It is easy verify that T−1 = T∗. Inother words, the inverse of T is the same as its adjoint (thetranspose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix T such that T−1AT = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental MatricesThis process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian, then the determination of T−1 is very simple.The eigenvectors v(1), ..., v(n) of A are known to be mutuallyorthogonal, so let us choose them so that they are also normalizedby < v(i), v(i) >= 1 for each i . It is easy verify that T−1 = T∗. Inother words, the inverse of T is the same as its adjoint (thetranspose of its complex conjugate).
Finally,
we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix T such that T−1AT = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental MatricesThis process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian, then the determination of T−1 is very simple.The eigenvectors v(1), ..., v(n) of A are known to be mutuallyorthogonal, so let us choose them so that they are also normalizedby < v(i), v(i) >= 1 for each i . It is easy verify that T−1 = T∗. Inother words, the inverse of T is the same as its adjoint (thetranspose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors,
then there is no matrix T such that T−1AT = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental MatricesThis process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian, then the determination of T−1 is very simple.The eigenvectors v(1), ..., v(n) of A are known to be mutuallyorthogonal, so let us choose them so that they are also normalizedby < v(i), v(i) >= 1 for each i . It is easy verify that T−1 = T∗. Inother words, the inverse of T is the same as its adjoint (thetranspose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix T
such that T−1AT = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental MatricesThis process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian, then the determination of T−1 is very simple.The eigenvectors v(1), ..., v(n) of A are known to be mutuallyorthogonal, so let us choose them so that they are also normalizedby < v(i), v(i) >= 1 for each i . It is easy verify that T−1 = T∗. Inother words, the inverse of T is the same as its adjoint (thetranspose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix T such that T−1AT = D. Inthis case,
A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Fundamental MatricesThis process is known as a similarity transformation.Alternatively, we may say that A is diagonalizable.
If A is Hermitian, then the determination of T−1 is very simple.The eigenvectors v(1), ..., v(n) of A are known to be mutuallyorthogonal, so let us choose them so that they are also normalizedby < v(i), v(i) >= 1 for each i . It is easy verify that T−1 = T∗. Inother words, the inverse of T is the same as its adjoint (thetranspose of its complex conjugate).
Finally, we note that if A has fewer than n linearly independenteigenvectors, then there is no matrix T such that T−1AT = D. Inthis case, A is not diagonalizable.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then λ is an eigenvalue of algebraic multiplicity 2 of the matrix A.In this event, there are two possibilities: The matrx A isnon-defectine and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then λ is an eigenvalue of algebraic multiplicity 2 of the matrix A.In this event, there are two possibilities: The matrx A isnon-defectine and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then λ is an eigenvalue of algebraic multiplicity 2 of the matrix A.In this event, there are two possibilities: The matrx A isnon-defectine and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues.
suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then λ is an eigenvalue of algebraic multiplicity 2 of the matrix A.In this event, there are two possibilities: The matrx A isnon-defectine and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation
∣∣∣A− λI∣∣∣ = 0
Then λ is an eigenvalue of algebraic multiplicity 2 of the matrix A.In this event, there are two possibilities: The matrx A isnon-defectine and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then λ is an eigenvalue of algebraic multiplicity 2 of the matrix A.In this event, there are two possibilities: The matrx A isnon-defectine and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then λ is an eigenvalue of algebraic multiplicity 2 of the matrix A.
In this event, there are two possibilities: The matrx A isnon-defectine and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then λ is an eigenvalue of algebraic multiplicity 2 of the matrix A.In this event, there are two possibilities:
The matrx A isnon-defectine and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then λ is an eigenvalue of algebraic multiplicity 2 of the matrix A.In this event, there are two possibilities: The matrx A isnon-defectine and
there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
We conclude our consideration of the linear homogeneous systemwith constant coefficients
x′ = Ax
with a discussion of the case in which the matrix A has a repeatedeigenvalues. suppose that λ is a repetead root of the characteristicequation ∣∣∣A− λI
∣∣∣ = 0
Then λ is an eigenvalue of algebraic multiplicity 2 of the matrix A.In this event, there are two possibilities: The matrx A isnon-defectine and there is still a fundamental set of solutions ofthe form
{veλt ,weλt
}.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation has a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
However,
if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation has a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
However, if the matrx A is defective,
there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation has a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution
ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation has a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt
associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation has a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue.
Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation has a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution,
it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation has a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation has a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0
when the characteristic equation has a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation has a doubleroot r .
In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation has a doubleroot r . In that case we found one exponential solution y1(t) = ert ,
but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation has a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation has a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way,
it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation has a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution
of the form
x = wteλt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation has a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
However, if the matrx A is defective, there is just one solution ofthe form veλt associated with this eigenvalue. Therefore, toconstruct the general solution, it is necessary to find other solutionof a different form.
Recall that a similar situation occurred for the linear equationay ′′ + by ′ + cy = 0 when the characteristic equation has a doubleroot r . In that case we found one exponential solution y1(t) = ert ,but a second independent solution had the form y2(t) = tert
In this way, it may be natural to attempt to find a secondindependent solution of the form
x = wteλt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
but,
doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
but, doing this and
substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
but, doing this and substituting x in the system
we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus,
we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and
substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system
we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and
only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
but, doing this and substituting x in the system we find thatw = 0. Thus, we propose
x = wteλt + ueλt
and substituting this new x in the system we find the system
(A− λI) w = 0
(A− λI) u = w
The first equation is already solved with w = v and only thesecond one is remaining to be solved.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Example 7.19
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Example 7.19
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Example 7.19
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Example 7.19
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Example 7.19
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Example 7.19
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Example 7.19
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Example 7.19
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Example 7.19
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Example 7.19
Find the solution of the system
x′ = Ax =
(1 −11 3
)x
Solution
Let’s find the eigenvalues of the matrix A
|A− λI| =
∣∣∣∣1− λ −11 3− λ
∣∣∣∣ = 0
(λ− 1)(λ− 3) + 1 = 0 =⇒ (λ− 2)2 = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
λ1 = 2,
λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)
and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
λ1 = 2, λ2 = 2,
If λ1,2 = 2, then
(A− λ1,2I) v =
(1− λ −1
1 3− λ
)(v1v2
)=
(−1 −11 1
)(v1v2
)=
(00
)and a corresponding eigenvector is
v(1) =
(1
− 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t +
ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u =
(A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u =
v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
and the solution is
x(1) =
(1
− 1
)e2t
Now, for the second solution we propose
x(2) = vte2t + ue2t
where u satisfies
(A− λI) u = (A− 2I) u = v
(−1 −11 1
)(u1u2
)=
(v1v2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + k
(1−1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + k
(1−1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + k
(1−1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k ,
where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + k
(1−1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary,
then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + k
(1−1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1.
If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + k
(1−1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + k
(1−1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + k
(1−1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + k
(1−1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + k
(1−1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)
then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + k
(1−1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + k
(1−1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + k
(1−1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + k
(1−1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t +
k
(1−1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
we have
−u1 − u2 = 1
so if u1 = k , where k is arbitrary, then u2 = −k − 1. If we write
u =
(k
−1− k
)=
(0−1
)+ k
(1−1
)then by substituting for w and u, we obtain
x(2) =
(1−1
)te2t +
(0−1
)e2t + k
(1−1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above
is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t)
and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but
the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation
shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) =
− e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0
and therefore{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}
form a fundamental set of solutions ofthe system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system.
The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x =
c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) +
c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
The last term above is merely a multiple of the first solutionx (1)(t) and may be ignored, but the first two terms constitute anew solution:
x(2) =
(1−1
)te2t +
(0−1
)e2t
An elementary calculation shows that W [x (1), x (2)](t) = − e4t 6= 0and therefore
{x (1), x (2)
}form a fundamental set of solutions of
the system. The general solution is
x = c1x(1) + c2x(2) = c1
(1−1
)e2t + c2
((1−1
)te2t +
(0−1
)e2t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but that thereis only one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but that thereis only one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but that thereis only one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A,
but that thereis only one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but that thereis only one corresponding eigenvector v.
Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but that thereis only one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but that thereis only one corresponding eigenvector v. Then one solution is
x(1)(t) = veλt
where v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but that thereis only one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but that thereis only one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but that thereis only one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but that thereis only one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but that thereis only one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = v
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the system
x′ = Ax
and suppose that r = λ is a double eigenvalue of A, but that thereis only one corresponding eigenvector v. Then one solution is
x(1)(t) = veλtwhere v satisfies
(A− λI) v = 0
and a second solution is given by
x(2)(t) = vteλt + ueλt
where u satisfies
(A− λI) u = vDr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Even though |A− λI| = 0,
it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u
( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .
Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation,
together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,
we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI)
[(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u =
(A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u =
0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Even though |A− λI| = 0, it can be shown that it is alwayspossible to solve it for u ( Actually, there are infinetly solutions ) .Now, Using the above equation, together with the equation for v,we get
(A− λI) [(A− λI) u = v]
(A− λI)2 u = (A− λI) v
(A− λI)2 u = 0
The vector u is known as a generalized eigenvector.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed
by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns.
Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix
forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19
can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) and
x (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t)
obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)=
e2t(
1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)
In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = I
can also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation
Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) =
Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)
Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0).
Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)
=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Fundamental Matrices
Fundamental matrices are formed by arranging linearly independentsolutions in columns. Thus, for example, a fundamental matrix forthe example 7.19 can be formed from the solutions x (1)(t) andx (2)(t) obtained before :
Ψ(t) =
(e2t te2t
−e2t −te2t − e2t
)= e2t
(1 t−1 −1− t
)In particular, the fundamental matrix Φ(t) that satisfies Φ(0) = Ican also be found from the relation Φ(t) = Ψ(t)Ψ−1(0). Thus, inthis case
Ψ(0) =
(1 0−1 −1
)=⇒ Ψ−1(0) =
(1 0−1 −1
)Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) =
Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) =
e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
)
(1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) =
e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)
The latter matrix is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix
is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as
the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A
can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if
it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of
n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors
(because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues),
then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed
into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called
its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Φ(t) = Ψ(t)Ψ−1(0) = e2t(
1 t−1 −1− t
) (1 0−1 −1
)
Φ(t) = e2t(
1− t −tt 1 + t
)The latter matrix is also known as the exponential matrix eAt .
Jordan Canonical Forms
An n × n matrix A can be diagonalized only if it has a fullcomplement of n linearly independent eigenvectors.
If there is a shortage of eigenvectors (because of repeatedeigenvalues), then A can always be transformed into a nearlydiagonal matrix called its Jordan form.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
A Jordan form, J,
has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
A Jordan form, J, has the eigenvalues of A
on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
A Jordan form, J, has the eigenvalues of A on the main diagonal,
ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and
zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3
. . .
λn 10 λn
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
A Jordan form, J, has the eigenvalues of A on the main diagonal,ones in certain positions on the diagonal above the main diagonal,and zeros elsewhere.
J(t) =
λ1 10 λ1 10 0 λ1
λ2 10 λ2
λ3. . .
λn 10 λn
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form,
we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T
with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v
in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and
the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column.
Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and
its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)
T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)
It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J =
T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
)
(1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
)
(1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Consider again the matrix A given by the equation
x′ = Ax =
(1 −11 3
)x
To transform A into its Jordan form, we construct thetransformation matrix T with the single eigenvector v in its firstcolumn and the generalized eigenvector u ( k = 0 ) in thesecond column. Then T and its inverse are given by
T =
(1 0−1 −1
)T−1 =
(1 0−1 −1
)It follows that
J = T−1AT =
(1 0−1 −1
) (1 −11 3
) (1 0−1 −1
)=
(2 10 2
)Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Ty where T is given above, produces thesystem
J′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Ty where T is given above, produces thesystem
J′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Ty where T is given above, produces thesystem
J′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Ty where T is given above, produces thesystem
J′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Ty
where T is given above, produces thesystem
J′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Ty where T is given above,
produces thesystem
J′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Ty where T is given above, produces thesystem
J′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Ty where T is given above, produces thesystem
J′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Ty where T is given above, produces thesystem
J′ = Jy
y ′1 = 2y1 + y2,
y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Ty where T is given above, produces thesystem
J′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Ty where T is given above, produces thesystem
J′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t ,
y1 = c1te2t + c2e
2t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Finally, If we start again from
x′ = Ax =
(1 −11 3
)x
the transformation x = Ty where T is given above, produces thesystem
J′ = Jy
y ′1 = 2y1 + y2, y ′2 = 2y2
y2 = c1e2t , y1 = c1te
2t + c2e2t
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus,
two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions
of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ;
y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)
Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I,
we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt .
To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix
for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system,
we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) =
TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)
which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same
as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix
that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7
Systems of First Order Linear EquationsSystems of First Order Linear Equations II
Homogeneous Linear Systems with Constant CoefficientsComplex EigenvaluesFundamental MatricesRepeated Eigenvalues
Repeated Eigenvalues
Thus, two independent solutions of the y−system are
y(1)(t) =
(10
)e2t ; y(2)(t) =
(t1
)e2t
and the corresponding fundamental matrix is
Ψ̂(t) =
(e2t te2t
0 e2t
)Since Ψ̂(0) = I, we can also identify this matrix as eJt . To obtaina fundamental matrix for the original system, we now form theproduct
Ψ(t) = TeJt =
(e2t te2t
−e2t −e2t − te2t
)which is the same as the fundamental matrix that we obtainedbefore.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 7