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Linear Algebra topicsL. Lanari
Control Systems
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 2
Matrices
M =
m11 m12 · · · m1q
m21 m22 · · · m2q...
... ·...
mp1 · · · mp,q−1 mpq
M = {mij : i = 1, . . . , p & j = 1, . . . , q}
MT =�m�
ij : m�ij = mji
�
1× 1
v : n× 1
vT : 1× n
M : p× p
M : p× q (p �= q)
transpose
vector (column)
row vector
scalar
square matrix
rectangular matrix
Terminologyand notation
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 3
Matrices
m11 m12 · · · m1n
0 m22. . . m2n
0. . .
. . ....
0 · · · 0 mnn
M1 0 00 M2 00 0 M3
�M11 M12
0 M22
�
M = diag{Mi}, i = 1, . . . , k
m1 0 · · · 0
0 m2. . . 0
0. . .
. . . 00 · · · 0 mn
M = diag{mi}, i = 1, . . . , pdiagonal matrix
block diagonal matrix
upper triangular matrix
upper block triangular matrix
lower triangular matrix
strictly lower triangular matrix
lower block triangular matrix
strictly lower block triangular matrix
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 4
Matrices
determinant of a diagonal matrix = product of the diagonal terms
det
�M11 M12
0 M22
�= det(M11) · det(M22)
det
�M1 00 M2
�= det(M1) · det(M2) special case
special case
useful for the eigenvalue computation
a11 · · · a1n...
. . ....
an1 · · · ann
λ1 · · · 0...
. . ....
0 · · · λn
=
λ1 · · · 0...
. . ....
0 · · · λn
a11 · · · a1n...
. . ....
an1 · · · ann
a11λ1 · · · a1nλn
.... . .
...an1λ1 · · · annλn
=
a11λ1 · · · a1nλ1
.... . .
...an1λn · · · annλn
gives
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 5
Matrices
det(M) �= 0
�A−1
�−1= A
(AB)−1 = B−1A−1
M = diag{mi} =⇒ M−1 = diag{ 1
mi}
non-singular(invertible)square matrix
A−1A = AA−1 = I
A0 = I
det�A�= det
�AT
�
⇔
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 6
generic transformationu
AuA
u
u
Au
Auor
particular directionssuch that Au is parallel to u
Au = λu
scalar(scaling factor)
Eigenvalues & Eigenvectors
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 7
Aui = λiui → (λiI −A)ui = 0
det(λiI −A) = 0
pA(λ) = det(λI −A) = λn + an−1λn−1 + an−2λ
n−2 + · · ·+ a1λ+ a0
λi
pA(λ) = det(λI −A) = 0
eigenvalue
for non-trivial solution
characteristic polynomial (order n = dim(A))
eigenvalue solution of
Eigenvalues & Eigenvectors
(scaling)
λi
eigenvector ui
ui �= 0
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 8
pA(λ) = det(λI −A) = λn + an−1λn−1 + an−2λ
n−2 + · · ·+ a1λ+ a0
λi
λi ∈ R
λi ∈ C λ∗i
ma(λi)
(λi,λ∗i )
generic eigenvalue
polynomial with real coefficients.
algebraicmultiplicity
Eigenvalues & Eigenvectors
λi ∈ Rλi ∈ C
ui
u∗iλ∗
i
real components
ui complex components
pA(λ) = 0
also is a solution pairs
therefore
if then
λieigenvalue pA(λ) = 0solution of with multiplicity
The set of the n solutions of is defined as the spectrum of A: ¾(A)
�
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 9
Eigenvalues & Eigenvectors
special cases
m11 m12 · · · m1n
0 m22. . . m2n
0. . .
. . ....
0 · · · 0 mnn
m1 0 · · · 0
0 m2. . . 0
0. . .
. . . 00 · · · 0 mn
diagonalmatrix
triangularmatrix
eigenvalues = {mi}
eigenvalues = {mii}
elements on themain diagonal
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 10
A TAT−1
Aui = λiui → ui
vTi A = vTi λi = λivTi → vTi
A(αui) = αAui = αλiui = λi(αui)
Eigenvalues & Eigenvectors
det(T ) �= 0
Tsame eigenvalues as A
right eigenvector
left eigenvector
eigenvalues are invariant under similarity transformations
eigenvector associated to ¸i is not unique
all belong to the same linear subspace
(proof)
vTi uj = δijwith
similarity transformations
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 11
Eigenvalues & Eigenvectors
¸i eigenvalue one or more linearly independent eigenvectors
A1 =
�λ1 00 λ1
�, u11 =
�10
�, u12 =
�01
�
A2 =
�λ1 β0 λ1
�, with β �= 0, only u1 =
�10
�
ma(λi) > 1
pA(λ) = (λ− λ1)2
Vi = {u ∈ Rn|Au = λiu}
Linear subspace (eigenspace) associated to an eigenvalue ¸i
dim(Vi) = mg(λi)
mg(λi) = dim [Ker(A− λiI)] = n− rank(A− λiI)
geometric multiplicity
with
A1 and A2 same eigenvalues with same m.a.
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 12
1 ≤ mg(λi) ≤ ma(λi) ≤ n
Eigenvalues & Eigenvectors
A1 =
�λ1 00 λ1
�, (A1 − λ1I) =
�0 00 0
�mg(λ1) = 2 = ma(λ1)
A2 =
�λ1 β0 λ1
�, (A2 − λ1I) =
�0 β0 0
�mg(λ1) = 1 < ma(λ1)
useful property
45°
u1
u2
u3
P =
0.5 0 0.50 1 00.5 0 0.5
u1 =
010
u2 =
101
u3 =
−101
¸1 = ¸2 = 1
¸3 = 0
projection matrix
ma(λ1) = 2 = mg(λ1)
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 13
Diagonalization
Def. An (n x n) matrix A is said to be diagonalizable if there exists an invertible (n x n) matrix T such that TAT -1 is a diagonal matrix
Th. An (n x n) matrix A is diagonalizable if and only if it has n independent eigenvectors
Since the eigenvalues are invariant under similarity transformations
if A diagonalizable
TAT−1 = Λ = diag{λi}, i = 1, . . . , n
eigenvalues of A
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 14
Diagonalization
Aui = λiui, i = 1, . . . , n
A�u1 u2 · · · un
�=
�u1 u2 · · · un
�
λ1 0 · · · 00 λ2 · 0
. . .0 · · · 0 λn
We need to find T
¸i ui n linearly independent(by hyp.)
non-singular
in matrix form
AU = UΛ
U =�u1 u2 · · · un
�
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 15
AU = UΛ
AT−1 = T−1Λ → A = T−1ΛT → Λ = TAT−1
Diagonalization
Distinct eigenvalues (real and/or complex) A diagonalizable=⇒⇐=
T−1 = Ubeing non-singular, we can define T such thatU
therefore the diagonalizing similarity transformation is T s.t.
T−1 = U =�u1 u2 · · · un
�
A: (n x n) is diagonalizable if and only if
mg(¸i) = ma(¸i) for every i
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 16
Diagonalization
Complex eigenvalues?λi = αi + jωi
ui = uai + jubi
eigenvalue
eigenvector
diagonalization
or real block 2 x 2
complexelements
realelements
(λi,λ∗i )
T−1 =�ui u∗
i
�→ Di = TAT−1 =
�λi 00 λ∗
i
�
T−1 =�uai ubi
�→ Mi = TAT−1 =
�αi ωi
−ωi αi
�
2 choices
real system representation for complex eigenvalues
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 17
Λr = diag{λ1, . . . ,λr}
Diagonalization
Simultaneous presence of real and complex eigenvalues
If A diagonalizable, there exists a non-singular matrix R such that
real eigenvalues
complex eigenvalues
RAR−1 = diag {Λr,Mr+1,Mr+3, . . . ,Mq−1}
Mi = TAT−1 =
�αi ωi
−ωi αi
�
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 18
Spectral decomposition or Eigendecomposition
A = U ΛU−1
U−1 =
vT1vT2...vTn
U−1U = I ⇒ vTi uj = δij , i, j = 1, . . . , n
A =n�
i=1
λi ui vTi
Hyp: A diagonalizable
U =�u1 u2 · · · un
�columns (linearly independent)
rows
A = U ΛU−1 =�λ1u1 λ2u2 · · · λnun
�
vT1vT2...vTn
spectral form of A
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 19
A =n�
i=1
λi ui vTi
Spectral decomposition
columnrow
is the projection matrix on the invariant subspace generated by ui
�(n x n) Pi = uiv
Ti
A =
�2 10 1
�u1 =
�10
�u2 =
�1−1
�
vT1 =�1 1
�vT2 =
�0 −1
�
P1 =
�1 10 0
�P2 =
�0 −10 1
�
λ1 = 2
λ2 = 1
u1
u2
P1 v
P2 v
v
v =
�−21
�
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 20
Not diagonalizable case
mg(λi) < ma(λi)
mg(λi) nkλi
Jk =
λi 1 0 · · · 00 λi 1 0...
. . .. . .
. . ....
0 · · · 0 λi 10 · · · · · · 0 λi
∈ Rnk×nk
blocks of dimension
Jordanblock
of dimension nk
Null space has dimension 1Jk − λiI
1 ≤ mg(λi) ≤ ma(λi) ≤ nSince in general
if A not diagonalizable then
the knowledge of this dimension is out of scope
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 21
Not diagonalizable case
Generalized eigenvector chain of nk generalized eigenvectors
Jordan canonical form (block diagonal)
ma(λi) = n =p�
i=1
nk
mg(λi) = p
example: unique eigenvalue ¸i of matrix A (n x n)
TAT−1 = J =
J1
. . .Jp
Jk ∈ Rnk×nk
∃ T :
Jordan block of dim nkwith
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 22
Special cases
• if ma(¸i) = 1 then mg(¸i) = 1
• if mg(¸i) = 1 then only one Jordan block of dimension ma(¸i)
then only one Jordan block of dimension ma(¸i)
�−1 10 −1
�λ1 = −1 ma(λ1) = 2 mg(λ1) = 1
From the rank-nullity theorem
rank(A - ¸iI)nullity(A - ¸iI)n
• if ¸i unique eigenvalue of matrix A and if rank(¸iI - A) = ma(¸i) - 1
dim (Rn) = dim (Ker(A− λiI)) + dim (Im(A− λiI))
A− λiI : Rn → Rn
Wednesday, October 8, 2014
Lanari: CS - Linear Algebra 23
Summary
A diagonalizable
A not diagonalizable
Λ = diag{λi}∃ T s.t. TAT−1 = Λ
mg(¸i) = ma(¸i) for all i
∃ T s.t.
TAT−1 = diag{Jk}
A =n�
i=1
λi ui vTi
Λr = diag{λ1, . . . ,λr}
Mi =
�αi ωi
−ωi αi
�
for real & complex ¸i
alternative choice for complex ¸i
Jk =
λi 1 · · · 0...
. . .. . .
...0 · · · λi 10 · · · 0 λi
∈ Rnk×nk
Jordan blocks
mg(¸i) < ma(¸i)
spectral form
block diagonal
Wednesday, October 8, 2014
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