Pseudospektren f¨ur strukturierte Matrixst¨orungen Michael ...... Basic question: How do the...

Pseudospektren fur strukturierte

Matrixstorungen

Michael Karow

Matheon, TU-Berlin

Eigenvalue perturbation theory.

An approach via pseudospectra and µ-functions

Michael Karow

Matheon, TU-Berlin

Motivation.

Situation:

The entries of a matrix A ∈ Cn×n have been obtained by measurement

A = A + ∆, ‖∆‖ ≤ ρ

↑ ↑ ↑ ↑| |measured value error bound

true value of A measurement error

Alternative point of view: Regard ∆ as a perturbation of A.

Problem: What is the spectrum of A ?

Well known:

The spectrum of a matrix can depend very sensitively on perturbations.

Example:

2 10k+1

10−k 5

︸︷︷︸Ak

=

2 10k+1

0 5

︸︷︷︸Ak

+

0 0

10−k 0

︸︷︷︸∆k

, k ∈ N

↓ ↓

Spectra: σ(Ak) = {0,7}, σ(Ak) = {2,5}

First order Taylor-Expansion of Eigenvalues of Ak yields the estimate

σ(Ak) ≈ {−1.3, 8.3}

The framework.

Basic question:

How do the eigenvalues of a matrix A change by adding matrix

perturbations ∆ of a certain class ∆ and norm ‖ · ‖ ?

A ; A + ∆, ∆ ∈ ∆, ‖∆‖ ≤ ρ.

More generally (Hinrichsen, Pritchard) :

How do the eigenvalues of A change under perturbations of the form

A ; A + B ∆C, ∆ ∈ ∆, ‖∆‖ ≤ ρ,

where B ∈ Cn×m and C ∈ Cp×n are fixed matrices ?

These are perturbations of feedback-form:

Open loop system: x = Ax + Bu, y = Cx

Feedback: u = ∆ y

Closed loop system: x = (A + B∆C)x ∆

(B,A,C)

Why B and C ?

These matrices can be used to model general linear perturbations of A

Examples:

•[A11 A12 + ∆A21 A22

]=

[A11 A12A21 A22

]+

[I0

]

︸︷︷︸B

∆[0 I

]

︸︷︷︸C

• A + (δ1 E1 + δ2 E2) = A +[I I

]

︸︷︷︸B

[δ1 I 00 δ2 I

] [E1E2

]

︸︷︷︸C

Question: How do the eigenvalues of A change under perturbations

of feedback form?

A ; A + B ∆C, ∆ ∈ ∆, ‖∆‖ ≤ ρ. (∗)

Related mathematical objects:

• Structured eigenvalue condition numbers

(they measure the change of eigenvalues for small perturbations)

• Spectral value sets (structured pseudospectra)

(Sets of all eigenvalues of all matrices of the form (∗))

• Stability radii

(smallest norm of a perturbation ∆ such that (∗) becomes unstable)

All these quantities can be computed via

• µ-values

• the structured distances of matrices to the set of singular matrices

(µ-values)

Definition of µ-values: For M ∈ Cm×n, ∆ ⊆ Cn×m,

µ∆(M) :=

(inf { ‖∆‖; ∆ ∈ ∆, det(∆M − In) = 0 }

)−1

.

In words:

1/µ∆(M) is the smallest norm of ∆ ∈ ∆ s.t. 1 is an eigenvalue of ∆M .

Easy to see:

∆ = Cn×n, ‖ · ‖ spectral norm ⇒ µ∆(M) = ‖M‖ = σmax(M).

The quantity µ∆(M) (the structured singular value) has been introduced by Doyle

and Safonov in the 1980’ies as a tool for robustness analysis of linear systems.

(; µ-Toolbox of MATLAB)

e w

w1 1

e

22

∆

M

Definition of µ again: For M ∈ Cm×n, ∆ ⊆ Cn×m,

µ∆(M) :=

(inf{ ‖∆‖; ∆ ∈ ∆, 1 ∈ σ(∆M) }

)−1

.

Main lemma:

For s ∈ C\σ(A) let G(s) = C(s − A)−1B. Then for all ∆ ∈ Cn×m,

1 ∈ σ(∆G(s)) ⇔ s ∈ σ(A + B∆C),

and hence

1

µ∆(G(s))= inf { ‖∆‖ ; ∆ ∈ ∆, s ∈ σ(A + B∆C) }

s

Eigenvalues of A

Eigenvalue of A+B C∆

Basic equivalence: For s 6∈ σ(A),

1 ∈ σ(∆G(s)) ⇔ s ∈ σ(A + B∆C).

Proof:

s ∈ σ(A + B∆C) ⇔ 0 = det(A − s + B∆C)

= det((A − s)(I − (s − A)−1B∆C))

⇔ 0 = det(I − (s − A)−1B∆C)

⇔ 1 ∈ σ((s − A)−1B∆C)

⇔ 1 ∈ σ(∆C(s − A)−1B︸︷︷︸G(s)

).

Application of µ: Stability radii

Definition: Stability radius with respect to open stability region Cg:

r∆(A, B, C, Cg) := inf { ‖∆‖; ∆ ∈ ∆, σ(A + B∆C) 6⊂ Cg }.

Formula for computation: If σ(A) ⊂ Cg then

r∆(A, B, C, Cg) = infs∈∂Cg

1

µ∆(G(s))=

1

sups∈∂Cgµ∆(G(s))

.

Proof: This is an immediate consequence of the main lemma

1

µ∆(G(s))= inf { ‖∆‖ ; ∆ ∈ ∆, s ∈ σ(A + B∆C) }.

� � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � ��

s

Eigenvalues of A

boundary of stability region

Application of µ: Spectral value sets (structured Pseudospectra)

Definition: Let (A, B, C) ∈ Cn×n × Cn×p × Cq×n.

σ∆(A, B, C, ρ) := Set of eigenvalues of all matrices of the form

A + B ∆C, ∆ ∈ ∆, ‖∆‖ ≤ ρ.

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

We have (see e.g. Hinrichsen and Pritchard, 2005)

σ∆(A, B, C, ρ) = σ(A) ∪ { s ∈ C; µ∆( C(sIn − A)−1B ) ≥ ρ−1 }.

Spectral value sets: The case B = C = In

Definition: Let A ∈ Cn×n.

σ∆(A, ρ) := Set of eigenvalues of all matrices of the form

A + ∆, ∆ ∈ ∆, ‖∆‖ ≤ ρ.

We have

σ∆(A, ρ) = { s ∈ C; µ∆( sIn − A ) ≤ ρ },

where

µ∆(A) := inf{ ‖∆‖; ∆ ∈ ∆, det(A − ∆) = 0 }

is the structured distance of A to the set of singular matrices.

For nonsingular A: µ∆(A) = 1/µ∆(A−1).

Useful characterizations of µ(M)

are known for the following cases

1. unstructured complex perturbations and operator norms

2. unstructured real perturbations and spectral norm

3. M complex vector, ∆ real vector, ‖∆‖ arbitrary

4. M block diagonal and ‖∆‖ = ‖ [ ‖∆jk‖ ] ‖.(coupled linear systems)

3. block diagonal complex perturbations and spectral norm

5. complex 1-parameter perturbations

6. M arbitrary ∆ ∈ Cn×n self- or skew-adjoint with respect to

an inner product.

Special case: ∆ complex hamiltonian.

Unstructured Complex µ for Operator Norms

Definition of µC:

µC(M) := 1/min{ ‖∆‖ | ∆ ∈ C`×q, 1 ∈ σ(∆M) }

Lemma: (well known)

‖∆‖ = ‖∆‖α,β = maxx6=0

‖∆x‖β

‖x‖α⇒ µC(M) = ‖M‖β,α = max

x6=0

‖Mx‖α

‖x‖β

Real Perturbations.

µ with respect to spectral norm

Complex Case: Real Case:

(well known) (Bernhardsson, Davisson, Doyle,Qiu, Rantzer, Young, ca. 1993)

µC(M) = σ1(M) µR(M) = infγ∈(0,1]

σ2

([<M −γ =M

γ−1=M <M

])

If M ∈ Cn×n then :

µC(M) = σn(M) µR(M) = supγ∈(0,1]

σ2n−1

([<M −γ =M

γ−1=M <M

])

Pseudospectra Example

The figures show real and complex pseudospectra of a real 4× 4 matrix.

pert.level: complex case: real case:

ρ = .05

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10




ρ = .08

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10




ρ = .098

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10




ρ = .1

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10




ρ = .15

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10




ρ = .19

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10




ρ = .2

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10




ρ = .3

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

Some real Spectral value sets

σR(A, B, C, ρ) = σ(A) ∪ { s ∈ C | µR(G(s)) ≥ ρ−1 }.

−3 −2 −1 0 1 2 3 4 5

−4

−3

−2

−1

0

1

2

3

4

G1(s) := g(s) diag(1, 14),

g(s) :=((s+1

2)2+1)((s+2)2+1)

s((s−1)2+1)((s+1)2+4),

−3 −2 −1 0 1 2 3 4 5

−4

−3

−2

−1

0

1

2

3

4

G2(s) := g(s) diag(1,0),

−3 −2 −1 0 1 2 3 4 5

−4

−3

−2

−1

0

1

2

3

4

G3(s) := diag( g(s), h(s) ),

h(s) :=((s+1

2)2+1)((s+2)2+1)+40

s((s−1)2+1)((s+1)2+4)

Example: Root sets of an interval polynomial

Center polynomial: p(s) = s12 −11∑

k=0

aksk =12∏

j=1

(s + j)

The figures show the sets

{ s ∈ C | sn −∑n−1k=0(ak + ∆k)s

k = 0, ∆k ∈ F, |∆k| ≤ 7 }

real perturbations complex perturbations

−50 −45 −40 −35 −30 −25 −20 −15 −10 −5 0

−30

−20

−10

0

10

20

30

−50 −45 −40 −35 −30 −25 −20 −15 −10 −5 0

−30

−20

−10

0

10

20

30

Hamiltonian Perturbations.

Definitions:

J-matrix: J =

[0 In

−In 0

]∈ C2n×2n

Fundamental property: J∗ = −J.

Set of complex ∗-Hamiltonian matrices:

Ham = {H ∈ C2n×2n; H∗J = −JH } (J-skew-adjoint matrices)

= {H ∈ C2n×2n; (JH)∗ = JH }

= { JH0 ∈ C2n×2n; H∗

0 = H0 }

=

{[A BC −A∗

]; B = B∗, C = C∗

}

Spectral symmetry:

The spectrum of a Hamiltonian matrix is

symmetric with respect to the imaginary axis:

λ ∈ σ(H) ⇒ −λ ∈ σ(H)� �

� ��

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� �

� ��

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� � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � �

i α

λ λ

imaginary Axis

0

µ for Hamiltonian Perturbations (K., 2007, submitted)

Hamiltonian matrices:

Ham = { ∆ ∈ C2n×2n; ∆J = (∆J)∗ }, J =

[0 I−I 0

]

Formulas for µ:

For M ∈ C2n×2n let H0 = M∗M, H1 = (JM)−(JM)∗2i . Then

µHam(M) = 1/min{ ‖∆‖2 ; 1 ∈ σ(∆M), ∆ ∈ Ham }

=√

mint∈R

λmax (H0 + t H1),

µHam(M) = min{ ‖∆‖2 ; det(M + ∆) = 0, ∆ ∈ Ham }.

=√

maxt∈R

λmin (H0 + t H1),

Example: Pseudospectra σ∆(H, ρ) for the hamiltonian matrix

H =

[A BC −A∗

],

B=diag(1,6,-6),

C=diag(-1,-6,6),A = 0.

Eigenvalues of H: ±i and ±6i (double).

∆ = Cn×n ∆=set of complex(unstr. perturbations) hamiltonian matrices

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

perturbation

level: ρ = 0.6


H =

[A BC −A∗

],

B=diag(1,6,-6),

C=diag(-1,-6,6),A = 0.



−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

perturbation

level: ρ = 1.0


H =

[A BC −A∗

],

B=diag(1,6,-6),

C=diag(-1,-6,6),A = 0.



−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

perturbation

level: ρ = 1.2


H =

[A BC −A∗

],

B=diag(1,6,-6),

C=diag(-1,-6,6),A = 0.



−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

perturbation

level: ρ = 1.6


H =

[A BC −A∗

],

B=diag(1,6,-6),

C=diag(-1,-6,6),A = 0.



−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

perturbation

level: ρ = 2.0


H =

[A BC −A∗

],

B=diag(1,6,-6),

C=diag(-1,-6,6),A = 0.



−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

perturbation

level: ρ = 3.0


H =

[A BC −A∗

],

B=diag(1,6,-6),

C=diag(-1,-6,6),A = 0.



−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

perturbation

level: ρ = 3.5


H =

[A BC −A∗

],

B=diag(1,6,6),

C=diag(-1,-6,-6),A = 0.



−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

perturbation

level: ρ = 3.5

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

Removing double eigenvalue from the imaginary axis

by a small perturbation

Let iω ∈ iR be a double eigenvalue of H ∈ Ham.

Then the following statements are equivalent.

(1) To each small ε > 0 there exists a Hamiltonian perturbation ∆ε ∈ Ham

such that ‖∆ε‖ ≤ ε and

σ(H + ∆ε) ∩ i [ω − ε, ω + ε] = ∅.

(2) There exists an eigenvector x 6= 0 such that

Hx = (iω) x, and x∗Jx = 0.

If (2) holds then

∆ε = εxx∗ + Jxx∗J

2 ‖x‖2is Hamiltonian, and

iω ± ε

2∈ σ(H + ∆ε).

Small Perturbations:

Pseudospectra and eigenvalue condition numbers.

The unstructured condition number of a simple eigenvalue

Let λ0 ∈ C be a simple eigenvalue of A∈Cn×n.

Let λ(A + ∆) be the eigenvalue of A + ∆ next to λ0. Then

cond(A, λ0) := limρ↘0

sup‖∆‖ ≤ ρ

0 6= ∆ ∈ Cn×n

|λ(A + ∆) − λ0|‖∆‖

Recall: Simple eigenvalues are differentiable:

λ(A + ∆) = λ0 +y∗∆x

y∗x+ O(‖∆‖2) where Ax = λ0 x, y∗A = λ0 y∗.

⇒cond(A, λ0) = Operator norm of the differential ∆ 7→

y∗∆x

y∗x

Problem:

Cond.numbers for multiple eigenvalues, structured perturbations.

Connection to pseudospectra.

ρ = .10

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

ρ = .07

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

ρ = .02

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

ρ = .10

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

zoom factor = ρ−1

ρ = .07

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−−−−−−− >

ρ = .02

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−−−−−−− >

ρ = .10

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

zoom factor = ρ−1

ρ = .07

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

4 4.5 5 5.5 6 6.5 7 7.5 8

2

2.5

3

3.5

4

4.5

5

5.5

6

−−−−−− >

ρ = .02

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

5.4 5.6 5.8 6 6.2 6.43.4

3.6

3.8

4

4.2

4.4

−−−−−− >

Example: The sets

σ(A, B, C, ρ) =⋃

∆ real, ‖∆‖2≤ρ

σ

([<(λ In + M) + ∆ −=(λ In + M)

=(λ In + M) <(λ In + M)

])

where λ = i and M =

[0 Z0 0

], Z =

[−2 + 2i −3 + 3i−2 + 3i −4 + 5i

]

ρ = 0.5 ρ = 0.2 ρ = 0.1 ρ = 0.01

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

↓ zoom ρ−1/2 ↓ zoom ρ−1/2

−1 −0.5 0 0.5 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

0.7

0.8

0.9

1

1.1

1.2

1.3

In the examples we have seen:

If properly scaled the connected components of

the spectral value sets σ∆(A, B, C, ρ) converge

to limit sets Lλ as ρ → 0.

Questions:

1. What is the right scaling?

2. How can the limit sets be computed?

To answer these questions one needs the

partial fraction decomposition of G(s) = C(sI − A)−1B

Partial Fraction Expansion of Transfer Function

Jordan decomposition: A =∑

λ∈σ(A)

(λ Pλ + Nλ )

Partial fraction expansion of resolvent (iλ = index of nilpotency):

(sIn − A)−1 =∑

λ∈σ(A)

Pλ

s − λ+

iλ∑

k=2

Nk−1λ

(s − λ)k

.

Hence, we have for the transfer function G(s) = C(sI − A)−1B:

G(s) =∑

λ∈σ(A)

CPλB

s − λ+

kλ∑

k=2

CNk−1λ B

(s − λ)k

Notation for leading coefficients:

0 6= Γλ :=

CPλB if kλ = 1,

CNkλ−1λ B otherwise.

Main result (K., in preparation)

Let Cλ(ρ) denote the connected component of σ∆(A, B, C, ρ)

that contains the eigenvalue λ ∈ σ(A).

Then we have with respect to the Hausdorff metric,

limρ↘0

Cλ(ρ) − λ

ρ1/kλ= Lλ,

where

Lλ := { s ∈ C; skλ ∈ σ(∆Γλ) for some ∆ ∈ ∆ with ‖∆‖ ≤ 1 }

= set of roots of order kλ of

the eigenvalues of ∆Γλ, ∆ ∈ ∆ with ‖∆‖ ≤ 1

= { r eiφ | φ∈ [0,2π), 0 ≤ r ≤ Rλ(φ) },where

Rλ(φ) := [µ∆(e−ikλφΓλ) ]1/kλ.

Radius of limit set equals the condition number

−5 0 5

−5

−4

−3

−2

−1

0

1

2

3

4

5

Limit Set

Condition Number

For small ρ,

Cλ(ρ) ≈ λ + ρ1/kλ Lλ

⇒

condition number := max{ |s|; s ∈ Lλ } = maxφ∈[0,2π]

[µ∆(e−ikλφΓλ) ]1/kλ

For an approach to condition numbers via Puisseux-series see paper by

Kressner, Moro, Pelaez (2006).

Example: Let 0 6= M ∈Cn×n be nilpotent and λ ∈ C \ R. Define

A :=

[<(λ In + M) −=(λ In + M)=(λ In + M) <(λ In + M)

]∈ R

2n×2n,

= S

[λ In + M 0

0 λ In + M

]S−1, where S =

1√2

[In In

−i In i In

].

Jordan decomposition: A = (λ Pλ + Nλ) + (λ Pλ + Nλ), where

Pλ = S

[In 00 0

]S−1, Nλ = S

[M 00 0

]S−1.

We now consider real perturbations of A of the form

A ; A∆ =

[<(λ In + M) + ∆ −=(λ In + M)

=(λ In + M) <(λ In + M)

]= A +

[In

0

]

︸︷︷︸B

∆[In 0

]

︸︷︷︸C

.

Leading coefficient in the part. fraction expansion of G(s):

Γλ = M i−1, i = index of nilpotency of M

Example continued: The sets

σ(A, B, C, ρ) =⋃

∆ real, ‖∆‖2≤ρ

σ

([<(λ In + M) + ∆ −=(λ In + M)

=(λ In + M) <(λ In + M)

])

where M =

[0 Z0 0

], Z =

[−2 + 2i −3 + 3i−2 + 3i −4 + 5i

]

ρ = 0.5 ρ = 0.2 ρ = 0.1 ρ = 0.01

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

↓ zoom ρ−1/2 ↓ zoom ρ−1/2

−1 −0.5 0 0.5 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

0.7

0.8

0.9

1

1.1

1.2

1.3

Example continued: The figures below show some limit sets

Lλ = { s ∈ C; si ∈ σ(∆Γλ) for some ∆ ∈ Rn×n with ‖∆‖2 ≤ 1 }, Γλ = M i−1

for M =

[0 Z0 0

](upper row) and M =

0 Z 00 0 Z0 0 0

(lower row)

where Z ∈ C2×2 is chosen at random.

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

−5 0 5−5

−4

−3

−2

−1

0

1

2

3

4

5

−5 0 5

−5

−4

−3

−2

−1

0

1

2

3

4

5

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4

Some limit sets for semisimple nonreal eigenvalues

Jordan decompositon: A = λ Pλ + λ Pλ + . . . , λ ∈ C \ R.

Limit sets:

Lλ = {s ∈ C; s ∈ σ(∆Pλ) for some ∆ ∈ Rn×n, ‖∆‖2 ≤ 1 }.

The figures show some limit sets for the case Pλ ∈ C4×4, rank(Pλ) = 2.

−10 −5 0 5 10

−10

−5

0

5

10

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

−2 −1 0 1 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

Limit sets for REAL perturbations of REAL eigenvalues

Consider A = (λ Pλ + Nλ) +∑

ν∈σ(A)\λ(ν Pν + Nν), λ, Pλ, Nλ real.

Perturbations: A ; A + ∆, ∆ real.

Let Nλ = N1, N2, N3, where

N1 =

0 1 00 0 10 0 0

, N2 =

0 2 00 0 20 0 0

, N3 = diag(N1, N2).

Limit sets:

−5 0 5−5

0

5

N1

−5 0 5−5

0

5

N2

−5 0 5−5

0

5

N3

Reason:

µR(e−3iφN2λ) =

σ1(N2λ) if e3iφ real

√σ1(N

2λ)σ2(N

2λ) otherwise.

REAL pseudospectra of a REAL 3 × 3 Jordan block

(w.r.t spectral norm)

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

Perturbation level ρ = 5 ∗ 101



−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Perturbation level ρ = 5 ∗ 10−1



−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4




−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2




−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08




−0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04


Radius of the pseudospectra of a Jordan block

−0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04Shape for small ρ

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

ρ+1

ρ1/3

Radius(ρ)

Perturbation Level ρ

Radius of Pseudospectum of 3x3 Jordan block

−50 −40 −30 −20 −10 0 10 20 30 40 50

−50

−40

−30

−20

−10

0

10

20

30

40

50

Shape for large ρ

For an n × n Jordan block:

limρ→0

Radius(ρ)

ρ1/n= 1, lim

ρ→∞(Radius(ρ) − ρ) = 1

Thanks for listening

−5 0 5−5

−4

−3

−2

−1

0

1

2

3

4

5

−5 0 5

−5

−4

−3

−2

−1

0

1

2

3

4

5

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4

Pseudospektren f¨ur strukturierte Matrixst¨orungen Michael ...... Basic question: How do the...

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