Post on 14-Oct-2020
a t ic a s
a t e’
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 2 / 26
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 3 / 26
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 3 / 26
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 3 / 26
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 3 / 26
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 3 / 26
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 3 / 26
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 3 / 26
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 3 / 26
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 3 / 26
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.
Then
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
(∑k
i=0 λ iBi
)
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 4 / 26
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.
Then
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
(∑k
i=0 λ iBi
)
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 4 / 26
Introduction E-val expansions: Regular case
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).
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 5 / 26
Introduction E-val expansions?: Singular case
E-vals are discontinuous functions of the entries
Example: Let us replace A0 +λA1 =[λ
000
]with the perturbed pencil
A0 +λ A1 =
[λ 00 0
]+ ε
([6 −3
−10 0
]+λ
[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!!
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 6 / 26
Introduction E-val expansions?: Singular case
E-vals are discontinuous functions of the entries
Example: Let us replace A0 +λA1 =[λ
000
]with the perturbed pencil
A0 +λ A1 =
[λ 00 0
]+ ε
([6 −3
−10 0
]+λ
[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!!
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 6 / 26
Introduction E-val expansions?: Singular case
E-vals are discontinuous functions of the entries
Example: Let us replace A0 +λA1 =[λ
000
]with the perturbed pencil
A0 +λ A1 =
[λ 00 0
]+ ε
([6 −3
−10 0
]+λ
[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!!
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 6 / 26
Introduction E-val expansions?: Singular case
...but they are continuous for most perturbations.
Wilkinson’s example (1978, LAA 1979):
A0 +λ A1 =
[λ 00 0
]+
([ε1 ε2ε3 ε4
]+λ
[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...
......
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 7 / 26
Introduction E-val expansions?: Singular case
...but they are continuous for most perturbations.
Wilkinson’s example (1978, LAA 1979):
A0 +λ A1 =
[λ 00 0
]+
([ε1 ε2ε3 ε4
]+λ
[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...
......
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 7 / 26
Introduction E-val expansions?: Singular case
...but they are continuous for most perturbations.
Wilkinson’s example (1978, LAA 1979):
A0 +λ A1 =
[λ 00 0
]+
([ε1 ε2ε3 ε4
]+λ
[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...
......
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 7 / 26
Introduction E-val expansions?: Singular case
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.
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 8 / 26
Introduction E-val expansions?: Singular case
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.
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 8 / 26
Introduction E-val expansions?: Singular case
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.
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 8 / 26
When do e-val expansions exist?
1 Introduction
2 When do e-val expansions exist?
3 First order term of the e-val expansions
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 9 / 26
When do e-val expansions exist?
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 10 / 26
When do e-val expansions exist?
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 11 / 26
When do e-val expansions exist?
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
P(λ )(A(λ )+ εB(λ ))Q(λ ) ≡
[D(λ ) 0
0 0
]+ ε
[G11(λ ) G12(λ )G21(λ ) G22(λ )
]
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 11 / 26
When do e-val expansions exist?
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
det(A(λ )+ εB(λ )) = C(λ ) det[
D(λ )+ εG11(λ ) εG12(λ )εG21(λ ) εG22(λ )
], C(λ0) 6= 0
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 11 / 26
When do e-val expansions exist?
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
det(A(λ )+ εB(λ )) = C(λ )εd det[
D(λ )+ εG11(λ ) G12(λ )εG21(λ ) G22(λ )
], C(λ0) 6= 0
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 11 / 26
When do e-val expansions exist?
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 11 / 26
When do e-val expansions exist?
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 11 / 26
When do e-val expansions exist?
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 11 / 26
When do e-val expansions exist?
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.
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 12 / 26
When do e-val expansions exist?
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 )
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 12 / 26
First order term of the e-val expansions First order term for simple e-vals
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 13 / 26
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)
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 14 / 26
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)
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 14 / 26
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)
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 14 / 26
First order term of the e-val expansions First order term for simple e-vals
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 ?
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 15 / 26
First order term of the e-val expansions First order term for simple e-vals
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 ?
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 15 / 26
First order term of the e-val expansions First order term for simple e-vals
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 ?
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 15 / 26
First order term of the e-val expansions First order term for simple e-vals
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 16 / 26
First order term of the e-val expansions First order term for simple e-vals
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)
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 17 / 26
First order term of the e-val expansions First order term for simple e-vals
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)
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 18 / 26
First order term of the e-val expansions First order term for simple e-vals
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)
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 18 / 26
First order term of the e-val expansions First order term for simple e-vals
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)
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 18 / 26
First order term of the e-val expansions First order term for simple e-vals
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)
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 18 / 26
First order term of the e-val expansions First order term for simple e-vals
The general (analytic) case: First order coefficient
A(λ ) analytic at λ0
Local Smith form at λ0: P(λ )A(λ )Q(λ ) =
[D(λ ) 0
0 0d×d
]
(2) Y HB(λ0)X +ζ Y HA′(λ0)X
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).
det(Y H1 B(λ0)X1)) 6= 0⇒(2) 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)
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 19 / 26
First order term of the e-val expansions First order term for simple e-vals
The general (analytic) case: First order coefficient
A(λ ) analytic at λ0
Local Smith form at λ0:[
P1(λ)P2(λ)
]A(λ )[ Q1(λ) Q2(λ) ] =
[D(λ ) 0
0 0d×d
]
(2) Y HB(λ0)X +ζ Y HA′(λ0)X
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).
det(Y H1 B(λ0)X1)) 6= 0⇒(2) 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)
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 19 / 26
First order term of the e-val expansions First order term for simple e-vals
The general (analytic) case: First order coefficient
A(λ ) analytic at λ0
Local Smith form at λ0:[
P1(λ)P2(λ)
]A(λ )[ Q1(λ) Q2(λ) ] =
[D(λ ) 0
0 0d×d
]
(2) Y HB(λ0)X +ζ Y HA′(λ0)X
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).
det(Y H1 B(λ0)X1)) 6= 0⇒(2) 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)
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 19 / 26
First order term of the e-val expansions First order term for simple e-vals
The general (analytic) case: First order coefficient
A(λ ) analytic at λ0
Local Smith form at λ0:[
P1(λ)P2(λ)
]A(λ )[ Q1(λ) Q2(λ) ] =
[D(λ ) 0
0 0d×d
]
(2) Y HB(λ0)X +ζ Y HA′(λ0)X
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).
det(Y H1 B(λ0)X1)) 6= 0⇒(2) 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)
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 19 / 26
First order term of the e-val expansions First order term for simple e-vals
The general (analytic) case: First order coefficient
A(λ ) analytic at λ0
Local Smith form at λ0:[
P1(λ)P2(λ)
]A(λ )[ Q1(λ) Q2(λ) ] =
[D(λ ) 0
0 0d×d
]
(2) Y HB(λ0)X +ζ Y HA′(λ0)X
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).
det(Y H1 B(λ0)X1)) 6= 0⇒(2) 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)
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 19 / 26
First order term of the e-val expansions First order term for simple e-vals
The simple case: regular vs singular
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 analytic matrices:
λ0(ε) = λ0 −?ε +O(ε2)
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 20 / 26
First order term of the e-val expansions First order term for simple e-vals
The simple case: regular vs singular
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 analytic matrices:
λ0(ε) = λ0 −det(Y HB(λ0)X )
(yHA′(λ0)x) ·det(Y H1 B(λ0)X1)
ε +O(ε2)
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 20 / 26
First order term of the e-val expansions First order term for multiple eigenvalues
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
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 21 / 26
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 ),
where
the mi different mi -th roots are considered, and
ζr , r = 1 : ri , are implicitly determined (solutions of a given equation).
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 22 / 26
First order term of the e-val expansions First order term for multiple eigenvalues
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
Set Φi = WiB(λ0)Zi , where
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 ),
where
the mi different mi -th roots are considered, and
ζr , r = 1 : ri , are the finite e-vals of a regular pencil.
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 23 / 26
First order term of the e-val expansions First order term for multiple eigenvalues
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
Set Φi = WiB(λ0)Zi , where
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 ),
where
the mi different mi -th roots are considered, and
ζr , r = 1 : ri , are the finite e-vals of a regular pencil.
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 23 / 26
First order term of the e-val expansions First order term for multiple eigenvalues
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)
H).
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 .
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 24 / 26
First order term of the e-val expansions First order term for multiple eigenvalues
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.
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 25 / 26
First order term of the e-val expansions First order term for multiple eigenvalues
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.
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 25 / 26
First order term of the e-val expansions First order term for multiple eigenvalues
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.
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 25 / 26
First order term of the e-val expansions First order term for multiple eigenvalues
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.
F. de Terán, F. M. Dopico and J. Moro (UC3M) 1st order spectral perturbation theory 2nd Najman Conference ’09 26 / 26