1st order spectral perturbation...

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First order spectral perturbation theory of squaresingular analytic matrix functions

Fernando de Terán

Departamento de MatemáticasUniversidad Carlos III de Madrid

(Spain)

2nd Najman Conference Joint work with F.M. Dopico and J.Moro

Dubrovnik

May, 2009

F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 1 / 26

Outline

1 IntroductionE-val expansions: Regular caseE-val expansions?: Singular case

2 When do e-val expansions exist?

3 First order term of the e-val expansionsFirst order term for simple e-valsFirst order term for multiple eigenvalues

Introduction

Basic Definitions

A(λ ), an n×n λ -dependent matrix (with complex coefficients) is,I Singular: if det A(λ ) ≡ 0

I Regular: otherwise

Normal Rank (nrank) of A(λ ): dimension of the largest non-indenticallyzero minor.

Eigenvalue (e-val) of A(λ ): µ∈ C such that: rank A(µ) < nrank A(λ ).

We will consider three types of matrices A(λ )

1 Matrix pencils: A(λ ) = A0 +λA1

2 Matrix polynomials: A(λ ) = A0 +λA1 + . . .+λ k Ak

3 A(λ ) analytic in a neighborhood of an eigenvalue λ0

Introduction

Basic Definitions

Introduction

Basic Definitions

Introduction

Basic Definitions

Introduction

Basic Definitions

Introduction

Basic Definitions

Introduction

Basic Definitions

Introduction

Basic Definitions

Introduction

Basic Definitions

Introduction E-val expansions: Regular case

Eigenvalue expansions for regular analytic matrices

A(λ ) regular, analytic near an eigenvalue λ0, with

detA(λ ) = (λ −λ0)aq(λ ) , q(λ ) analytic at λ0 and q(λ0) 6= 0

A(λ )+ εB(λ ) a perturbation of A(λ ), with B(λ ) analytic at λ = λ0.

There are a eigenvalues of A(λ )+ εB(λ ), λi(ε), i = 1, . . . ,a, with λi(0) = λ0,which can be expanded as (fractional) power series in ε

This include

1 Pencils: A(λ )+ εB(λ ) = A0 +λA1 + ε(B0 +λB1)

2 Polynomials: A(λ )+ εB(λ ) = ∑ki=0 λ iAi + ε

i=0 λ iBi

Eigenvalue expansions for regular analytic matrices

A(λ ) regular, analytic near an eigenvalue λ0, with

detA(λ ) = (λ −λ0)aq(λ ) , q(λ ) analytic at λ0 and q(λ0) 6= 0

A(λ )+ εB(λ ) a perturbation of A(λ ), with B(λ ) analytic at λ = λ0.

There are a eigenvalues of A(λ )+ εB(λ ), λi(ε), i = 1, . . . ,a, with λi(0) = λ0,which can be expanded as (fractional) power series in ε

This include

1 Pencils: A(λ )+ εB(λ ) = A0 +λA1 + ε(B0 +λB1)

2 Polynomials: A(λ )+ εB(λ ) = ∑ki=0 λ iAi + ε

i=0 λ iBi

First order expansions for regular analytic matrices

I Langer and Najman (1992): Describe the first order term of theeigenvalue expansions using the local Smith form of A(λ ) at λ0

I Lancaster, Markus and Zhou (2003): Describe the first order term of theeigenvalue expansions using bases of kerA(λ0) and kerA(λ0)

H (onlysemisimple eigenvalues).

Introduction E-val expansions?: Singular case

E-vals are discontinuous functions of the entries

Example: Let us replace A0 +λA1 =[λ

]with the perturbed pencil

A0 +λ A1 =

[λ 00 0

([6 −3

−10 0

[0 11 0

[λ + ε6 ε(λ −3)

ε(λ −10) 0

e-val of A0 +λA1 = {0}

e-vals of A0 +λ A1 = {3,10} for any ε 6= 0!!

A0 +λ A1 =

[λ 00 0

([6 −3

−10 0

[0 11 0

[λ + ε6 ε(λ −3)

ε(λ −10) 0

A0 +λ A1 =

[λ 00 0

([6 −3

−10 0

[0 11 0

[λ + ε6 ε(λ −3)

ε(λ −10) 0

...but they are continuous for most perturbations.

Wilkinson’s example (1978, LAA 1979):

A0 +λ A1 =

[λ 00 0

([ε1 ε2ε3 ε4

[η1 η2η3 η4

Wilkinson’s Comment: Although quite respectable e-vals may be completelydestroyed by arbitrarily small perturbations, for almost all small perturbationsεi and ηi , the pencil A0 +λ A1 has an e-val close to 0.

Numerical experiments: (A0 +λA1)+(E0 +λE1) =

[λ 00 0

]+ random ‖ · ‖ = 10−5

Case λ1 λ2

1 -0.000095 1.6343572 -0.000004 2.3368623 0.000027 1.1572664 0.000033 3.2407605 -0.000016 2.3897946 -0.000006 -0.300021...

......

A0 +λ A1 =

[λ 00 0

([ε1 ε2ε3 ε4

[η1 η2η3 η4

[λ 00 0

]+ random ‖ · ‖ = 10−5

Case λ1 λ2

1 -0.000095 1.6343572 -0.000004 2.3368623 0.000027 1.1572664 0.000033 3.2407605 -0.000016 2.3897946 -0.000006 -0.300021...

......

A0 +λ A1 =

[λ 00 0

([ε1 ε2ε3 ε4

[η1 η2η3 η4

[λ 00 0

]+ random ‖ · ‖ = 10−5

Case λ1 λ2

1 -0.000095 1.6343572 -0.000004 2.3368623 0.000027 1.1572664 0.000033 3.2407605 -0.000016 2.3897946 -0.000006 -0.300021...

......

Our Goals

To characterize generic perturbations of general square singular matrixfunctions A(λ ) for which e-vals change continuously.

For these perturbations to develop first order perturbation expansions forthe variation of e-vals (simple or multiple)

A property is said to be GENERIC if it holds for all perturbations except thosein a set of zero Lebesgue measure.

Our Goals

When do e-val expansions exist?

1 Introduction

3 First order term of the e-val expansions

Notation

Unperturbed matrix: A(λ ) analytic in a neighborhood of λ0 ∈ C

(SINGULAR)

Perturbation matrix: B(λ ) analytic in a neighborhood of λ0 ∈ C

Perturbed matrix: A(λ )+ εB(λ )

All with size n×n

Right null space of A(λ ) at µ: N (A(µ)) = {x ∈ Cn : A(µ)x = 0}

Left null space of A(λ ) at µ ≡ Right null space of A(λ )H at µ

Regularizing the problem

A(λ ) singular analytic near λ0.

Local Smith form at λ0: P(λ )A(λ )Q(λ ) =

[D(λ ) 0

0 0d×d

], where

D(λ ) = diag((λ −λ0)m1 , . . . ,(λ −λ0)

mg , I), with 0 < m1 ≤ . . . ≤ mg ,

P(λ ),Q(λ ) analytic and invertible near λ0

Notation: a = m1 + . . .+mg

If ε 6= 0, e-vals of A(λ )+ εB(λ ) near λ0 are the roots of

pε(λ ) = det[

D(λ )+ εG11(λ ) G12(λ )εG21(λ ) G22(λ )

]= (λ −λ0)

a detG22(λ )+ ε qε(λ ),

whose coefficients are polynomials in ε.

limε→0 pε(λ ) = (λ −λ0)a detG22(λ ) 6≡ 0 ⇔ detG22(λ ) 6≡ 0

[D(λ ) 0

0 0d×d

], where

D(λ ) = diag((λ −λ0)m1 , . . . ,(λ −λ0)

mg , I), with 0 < m1 ≤ . . . ≤ mg ,P(λ ),Q(λ ) analytic and invertible near λ0

P(λ )(A(λ )+ εB(λ ))Q(λ ) ≡

[D(λ ) 0

[G11(λ ) G12(λ )G21(λ ) G22(λ )

pε(λ ) = det[

D(λ )+ εG11(λ ) G12(λ )εG21(λ ) G22(λ )

]= (λ −λ0)

[D(λ ) 0

0 0d×d

], where

D(λ ) = diag((λ −λ0)m1 , . . . ,(λ −λ0)

det(A(λ )+ εB(λ )) = C(λ ) det[

D(λ )+ εG11(λ ) εG12(λ )εG21(λ ) εG22(λ )

], C(λ0) 6= 0

pε(λ ) = det[

D(λ )+ εG11(λ ) G12(λ )εG21(λ ) G22(λ )

]= (λ −λ0)

[D(λ ) 0

0 0d×d

], where

D(λ ) = diag((λ −λ0)m1 , . . . ,(λ −λ0)

det(A(λ )+ εB(λ )) = C(λ )εd det[

D(λ )+ εG11(λ ) G12(λ )εG21(λ ) G22(λ )

], C(λ0) 6= 0

pε(λ ) = det[

D(λ )+ εG11(λ ) G12(λ )εG21(λ ) G22(λ )

]= (λ −λ0)

[D(λ ) 0

0 0d×d

], where

D(λ ) = diag((λ −λ0)m1 , . . . ,(λ −λ0)

mg , I), with 0 < m1 ≤ . . . ≤ mg ,

pε(λ ) = det[

D(λ )+ εG11(λ ) G12(λ )εG21(λ ) G22(λ )

]= (λ −λ0)

[D(λ ) 0

0 0d×d

], where

D(λ ) = diag((λ −λ0)m1 , . . . ,(λ −λ0)

mg , I), with 0 < m1 ≤ . . . ≤ mg ,

pε(λ ) = det[

D(λ )+ εG11(λ ) G12(λ )εG21(λ ) G22(λ )

]= (λ −λ0)

[D(λ ) 0

0 0d×d

], where

D(λ ) = diag((λ −λ0)m1 , . . . ,(λ −λ0)

mg , I), with 0 < m1 ≤ . . . ≤ mg ,

pε(λ ) = det[

D(λ )+ εG11(λ ) G12(λ )εG21(λ ) G22(λ )

]= (λ −λ0)

The general existence Theorem

Theorem:If B(λ ) satisfies detG22(λ ) 6≡ 0 then

There exists K > 0 such that

A(λ )+ εB(λ ) is regular for 0 < |ε | < K .

There are a e-vals {λ1(ε), . . . ,λa(ε)} of A(λ )+ εB(λ ) which are(fractional) power series of ε and such that

limε→0

λi(ε) = λ0 for i = 1 : a.

The general existence Theorem

Theorem:

If B(λ ) satisfies detG22(λ ) 6≡ 0 then

There exists K > 0 such that

A(λ )+ εB(λ ) is regular for 0 < |ε | < K .

There are a e-vals {λ1(ε), . . . ,λa(ε)} of A(λ )+ εB(λ ) which are(fractional) power series of ε and such that

limε→0

λi(ε) = λ0 for i = 1 : a.

detG22(λ ) 6≡ 0 is generic in the set of perturbations B(λ )

First order term of the e-val expansions First order term for simple e-vals

The regular case

Matrix case: Ax = λ0x , yHA = λ0yH . There is a unique

e-val of A+ εB such that

λ0(ε) = λ0 +yHBxyHx

ε +O(ε2)

Regular analytic matrices: A(λ0)x = 0, yHA(λ0) = 0. There

is a unique e-val of A(λ )+ εB(λ ) such that

λ0(ε) = λ0 −yHB(λ0)xyHA′(λ0)x

ε +O(ε2)

Singular square analytic matrices:

λ0(ε) = λ0 −?ε +O(ε2)

The regular case

ε +O(ε2)

λ0(ε) = λ0 −?ε +O(ε2)

The regular case

ε +O(ε2)

λ0(ε) = λ0 −?ε +O(ε2)

The associated pencil

Theorem:λ0 simple e-val of A(λ ), X and Y bases of right and left null spaces of A(λ0).If the pencil

Y HB(λ0)X +ζ Y HA′(λ0)X

is regular and has only one finite eigenvalue, ζ0, there is a unique e-val ofA(λ )+ εB(λ )

λ (ε) = λ0 +ζ0ε +O(ε2).

Sufficient conditions?

Explicit description of ζ0 ?

is regular and has only one finite eigenvalue , ζ0, there is a unique e-val ofA(λ )+ εB(λ )

λ (ε) = λ0 +ζ0ε +O(ε2).

is regular and has only one finite eigenvalue , ζ0, there is a unique e-val ofA(λ )+ εB(λ )

λ (ε) = λ0 +ζ0ε +O(ε2).

The pencil case: Reducing subspaces

Reducing subspaces play in singular pencils the role of invariant subspaces inmatrices, and deflating subspaces in regular pencils.

N (A(λ )) = {x(λ ) ∈ Cn(λ ) : A(λ )x(λ ) ≡ 0}

(vector space over C(λ ) ={

p(λ )q(λ )

: p(λ ),q(λ ) ∈ C[λ ], q(λ ) 6≡ 0}

Definition (Van Dooren 1982):Let X ⊆ C

n be a subspace. X is a reducing subspace of the m×n pencilA0 +λA1 if dim(A0X +A1X ) = dim(X )−dimN (A0 +λA1)

For arbitrary subspaces Z ∈ Cn×n,

dim(A0Z +A1Z ) ≥ dim(Z )−dimN (A0 +λA1)

Minimal reducing subspace: R ⊆⋂

X r .s. X

The polynomial case: Singular subspaces

A(λ ) = A0 +λA1 + . . .+λ k Ak

Singular subspaces are the generalization of the minimal reducing subspacesto matrix polynomials.

N (A(λ )) = {x(λ ) ∈ Cn(λ ) : A(λ )x(λ ) ≡ 0}

Lemma-DefinitionGiven µ ∈ C, the subspace R(µ) := Span{v1(µ), . . . ,vd (µ)} is the same for allpolynomial bases {v1(λ ), . . . ,vd (λ )} of N (A(λ )) such that {v1(µ), . . . ,vd (µ)}is linearly independent.R(µ) : Right singular subspace of A(λ ) at µ

L (µ): Left singular subspace of A(λ ) at µ (Similar for left null vectors)

The pencil/polynomial case: First order coefficient

A(λ ) = A0 +λA1 + . . .+λ k Ak , B(λ ) = B0 +λB1 + . . .+λ k Bk

(1) Y HB(λ0)X +ζ Y HA′(λ0)X

R (resp. L ):{minimal reducing subspace of A0 +λA1 (resp. AH

0 +λAH1 ) (pencil)

right (resp. left) singular subspace of A(λ ) at λ0 (polynomial)

X1 (resp. Y1): basis of R ∩N (A(λ0)) (resp. L ∩N (A(λ0)H))

X = [x X1] (resp. Y = [y Y1]): basis of N (A(λ0) (resp. N (A(λ0)H)) (x ,y

vectors).

det(Y H1 B(λ0)X1)) 6= 0⇒(1) is regular and has only one finite e-val

Then: ζ0 = − det(Y HB(λ0)X )

(yHA′(λ0)x)·det(Y H1 B(λ0)X1)

0 +λAH1 ) (pencil)

vectors).

0 +λAH1 ) (pencil)

vectors).

0 +λAH1 ) (pencil)

vectors).

The general (analytic) case: First order coefficient

A(λ ) analytic at λ0

[D(λ ) 0

0 0d×d

Y H1 := P2(λ0)

H ∈ N (A(λ0)H) (resp. X1 := Q2(λ0) ∈ N (A(λ0)))

X = [x X1] (resp. Y = [y Y1]): basis of N (A(λ0)) (resp. N (A(λ0)H)) (x ,y

vectors).

Local Smith form at λ0:[

P1(λ)P2(λ)

]A(λ )[ Q1(λ) Q2(λ) ] =

[D(λ ) 0

0 0d×d

Y H1 := P2(λ0)

vectors).

P1(λ)P2(λ)

]A(λ )[ Q1(λ) Q2(λ) ] =

[D(λ ) 0

0 0d×d

Y H1 := P2(λ0)

vectors).

P1(λ)P2(λ)

]A(λ )[ Q1(λ) Q2(λ) ] =

[D(λ ) 0

0 0d×d

Y H1 := P2(λ0)

vectors).

P1(λ)P2(λ)

]A(λ )[ Q1(λ) Q2(λ) ] =

[D(λ ) 0

0 0d×d

Y H1 := P2(λ0)

vectors).

The simple case: regular vs singular

ε +O(ε2)

Singular analytic matrices:

λ0(ε) = λ0 −?ε +O(ε2)

The simple case: regular vs singular

ε +O(ε2)

Singular analytic matrices:

λ0(ε) = λ0 −det(Y HB(λ0)X )

(yHA′(λ0)x) ·det(Y H1 B(λ0)X1)

ε +O(ε2)

First order term of the e-val expansions First order term for multiple eigenvalues

Multiple eigenvalues: regular caseConsider a group of ri equal exponents in the local Smith of A(λ ) at λ0:

· · · < mi+1 = mi+2 = · · · = mi+ri < · · ·

Set: mi ≡mi+1 = mi+2 = · · · = mi+ri

Set Φi = WiB(λ0)Zi , where

Wi ≡ rows of P(λ0) corresponding to exponents≥ miZi ≡ columns of Q(λ0) corresponding to exponents≥ mi

Theorem (Langer & Najman, 1992)If detΦi+1 6= 0 there are ri mi e-vals of A(λ )+ εB(λ ) such that

λ (ε) = λ0 +ζ 1/mir ε1/mi +o(ε1/mi ),

the mi different mi -th roots are considered, and

ζr , r = 1 : ri , are implicitly determined (solutions of a given equation).

Multiple eigenvalues: singular case (I)Consider a group of ri equal exponents in the local Smith of A(λ) at λ0:

· · · < mi+1 = mi+2 = · · · = mi+ri < · · ·

Set: mi ≡mi+1 = mi+2 = · · · = mi+ri

Wi ≡ rows of P(λ0) corresponding to exponents≥ miand also the last d rows

Zi ≡ columns of Q(λ0) corresponding to exponents≥ miand also the last d columns

TheoremIf detΦi+1 6= 0 there are ri mi e-vals of A(λ )+ εB(λ ) such that

λ (ε) = λ0 +ζ 1/mir ε1/mi +o(ε1/mi ),

ζr , r = 1 : ri , are the finite e-vals of a regular pencil.

Multiple eigenvalues: singular case (I)Consider a group of ri equal exponents in the local Smith of A(λ) at λ0:

· · · < mi+1 = mi+2 = · · · = mi+ri < · · ·

Set: mi ≡mi+1 = mi+2 = · · · = mi+ri

Wi ≡ rows of P(λ0) corresponding to exponents≥ miand also the last d rows

Zi ≡ columns of Q(λ0) corresponding to exponents≥ miand also the last d columns

TheoremIf detΦi+1 6= 0 there are ri mi e-vals of A(λ )+ εB(λ ) such that

λ (ε) = λ0 +ζ 1/mir ε1/mi +o(ε1/mi ),

ζr , r = 1 : ri , are the finite e-vals of a regular pencil.

Multiple eigenvalues: singular case (II)

λ (ε) = λ0 +ζr1/mi +o(‖[E0 E1]‖

1/mi ), with ζr , r = 1 : ri , the finite eigenvalues ofa regular pencil.

This pencil is

Φi +ζ[

Iri 00 0

Φi is built up using particular bases, properly normalized, of N (A(λ0))and N (A(λ0)

Drawbacks on normalizations of bases are related to the multiple regularcase, not to singularity.

If λ0 is semisimple then arbitrary bases X ,Y can be chosen, and thepencil is Y HB(λ0)X +ζY HA′(λ0)X .

Conclusions

We have presented generic sets of perturbations of square singularmatrix functions, analytic in a neighborhood of a given eigenvalue, forwhich an eigenvalue perturbation theory is possible.

For perturbations in these sets, first order perturbation expansions ofsimple and multiple eigenvalues have been developed.

The particular cases of matrix pencils and matrix polynomials have beenaddressed. In these cases, the leading coefficients of the eigenvalueexpansions can be described using some spectral information of theunperturbed matrix function.

Conclusions

Bibliography

F. DE TERÁN, F. M. DOPICO AND J. MORO, First order spectralperturbation theory of square singular matrix pencils, Linear AlgebraAppl., 429 (2008) 548–576.

F. DE TERÁN AND F. M. DOPICO, First order spectral perturbation theoryof square singular matrix polynomials, submitted.

P. LANCASTER, A. S. MARKUS AND F. ZHOU, Perturbation theory foranalytic matrix functions: the semisimple case , SIAM J. Matrix Anal.Appl., 25 (2003) 606-626.

H. LANGER AND B. NAJMAN, Remarks on the perturbation of analyticmatrix functions III, Integr. Equat. Oper. Th., 15 (1992), pp. 796–806.

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