Control Systems - dis.uniroma1.itlanari/ControlSystems/CS - Lectures/2014_Lec_04... · Lanari: CS -...

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



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



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



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

�

⇔



generic transformationu

AuA

u

u

Au

Auor

particular directionssuch that Au is parallel to u

Au = λu

scalar(scaling factor)

Eigenvalues & Eigenvectors



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


(scaling)

λi

eigenvector ui

ui �= 0



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


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)

�




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



A TAT−1

Aui = λiui → ui

vTi A = vTi λi = λivTi → vTi

A(αui) = αAui = αλiui = λi(αui)


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




¸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.



1 ≤ mg(λi) ≤ ma(λi) ≤ n


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)



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



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

�



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



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



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

�



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



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

�



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



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



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



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


Control Systems - dis.uniroma1.itlanari/ControlSystems/CS - Lectures/2014_Lec_04... · Lanari: CS -...

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