linear algebra
DESCRIPTION
Linear Algebra. Lecture 36. Revision Lecture I. Seg V and III. Eigenvalues and Eigenvectors. If A is an n x n matrix, then a scalar is called an eigenvalue of A if there is a nonzero vector x such that Ax = x . …. - PowerPoint PPT PresentationTRANSCRIPT
If A is an n x n matrix, then a scalar is called an eigenvalue of A if there is a nonzero vector x such that Ax = x.
…
λ
λ
If A is a triangular matrix then the eigenvalues of A are the entries on the main diagonal of A.
If is an eigenvalue of a matrix A and x is a corresponding eigenvector, and if k is any positive integer, then is an eigenvalue of Ak and x is a corresponding eigenvector.
λ
kλ
det( ) 0A I
If A and B are n x n matrices, then A is similar to B if there is an invertible matrix P such that P -1AP = B, or equivalently, A = PBP -1.
…
If n x n matrices A and B are similar, then they have the same characteristic polynomial and hence the same Eigenvalues.
A square matrix A is said to be diagonalizable if A is similar to a diagonal matrix.
i.e. if A = PDP -1 for some invertible matrix P and some diagonal matrix D.
An n x n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors.
An n x n matrix with n distinct eigenvalues is diagonalizable.
Let V and W be n-dim and m-dim spaces, and T be a LT from V to W. To associate a matrix with T we chose bases B and C for V and W respectively …
Given any x in V, the coordinate vector [x]B is in Rn and the [T(x)]C coordinate vector of its image, is in Rm
…
Let {b1 ,…,bn} be the basis B for V.
If x = r1b1 +…+ rnbn, then
1
[ ]Bx
n
r
r
1
1
( ) ( )
( ) ( )n
n
r r
r r
and
1 n
1 n
T x T b b
T b T b …
Connection between [ x ]B and [T(x)]C
1[ ( )] [ ( )] [ ( )]C 1 C n CT x T b T b nr r
[ ( )] [ ]C BT x M x
[ ( )] [ ( )] [ ( )]
where1 C 2 C n CM T b T b T b
This equation can be written as
The Matrix M is the matrix representation of T, Called the matrix for T relative to the bases B and C
Similarity of two matrix representations: A=PCP-1
A complex scalar satisfies
if and only if there is a nonzero vector x in Cn such that We call a (complex) eigenvalue and x a (complex) eigenvector corresponding to .
λdet( - ) 0A I
λ
λ
.Ax x
x xr r
rB r Bx xB B
1 kAx kx
If A has two complex eigenvalues whose absolute value is greater than 1, then 0 is a repellor and iterates of x0 will spiral outward around the origin.
…
If the absolute values of the complex eigenvalues are less than 1, the origin is an attractor and the iterates of x0 spiral inward toward the origin.
x Ax1 1
11 1
1
( ) ( )( ) , ( ) ,
( ) ( )x x
A
n n
n
n nn
x t x tt t
x t x t
a a
a a
where
and
(0)
0
x Axx x
SolveSubject to
( ) tt ve
x Ax
x
For the generalequationSolution might be a linear combination of the form …
( )
( )
t
t
t e
t e
x v
Ax Av0,
( )
tet (t)
Since
iff ,i.e. iff λ is aneigen value ofand is a corresponding eigenvector.
x Ax v AvA
v…
( ) tt e of .x v x Ax
Thus each eigenvalue - eigenvector pair provides a solution Such solutions are sometimes called eigen functions of the differential equation.
3 x 3 Determinant
11 12 13
21 22 23
31 32 33
11 22 33 12 23 31 13 21 32
13 22 31 11 23 32 12 21 33
det( )a a a
A a a aa a a
a a a a a a a a aa a a a a a a a a
11 12 13
21 22 23
31 32 33
a a aA a a a
a a a
11 11 12 12
11 1
det det det
... ( 1) detnn n
A a A a A
a A
Expansion
11 1
1
( 1) detn
jj j
j
a A
Minor of a MatrixIf A is a square matrix, then the Minor of entry aij (called the ijth minor of A) is denoted by Mij and is defined to be the determinant of the sub matrix that remains when the ith row and jth column of A are deleted.
CofactorThe number Cij=(-1)i+j Mij is called the cofactor
of entry aij
(or the ijth cofactor of A).
Cofactor Expansion Across the First Row
11 11 12 12 1 1det ... n nA a C a C a C
( 1) deti ji j ijC A
The determinant of a matrix A can be computed by a cofactor expansion across any row or down any column.
The cofactor expansion across the ith row
1 1 2 2det ...i i i i in inA a C a C a C
The cofactor expansion down the jth column
1 1 2 2det ...j j j j nj njA a C a C a C
If A is triangular matrix, then det (A) is the product of the entries on the main diagonal.
11
21 22
31 32 33
41 42 43 44
0 0 00 0
0
aa a
Aa a aa a a a
11 22 33 44det( )A a a a a
Let A be a square matrix.
If a multiple of one row of A is added to another row to produce a matrix B, then det B = det A.
…..
ContinueIf two rows of A are
interchanged to produce B, then det B = –det A.If one row of A is multiplied
by k to produce B, then det B = k det A.
If A is an n x n matrix, thendet AT = det A.
If A and B are n x n matrices, then
det (AB)=(det A )(det B)
ObserveFor any n x n matrix A and any b in Rn, let Ai(b) be the matrix obtained from A by replacing column i by the vector b.
1( ) ... ...i nA b a b a
coli
Let A be an invertible n x n matrix. For any b in Rn, the unique solution x of Ax = b has entries given by
det ( ), 1,2,...,
deti
iA b
x i nA
Let A be an invertible matrix, then
1 1det
A adj AA
Let T: R2 R2 be the linear transformation determined by a 2 x 2 matrix A. If S is a parallelogram in R2, then{area of T (S)} = |detA|. {area of S}
If T is determined by a 3 x 3 matrix A, and if S is a parallelepiped in R3, then{volume of T (S)} = |detA|. {volume of S}