On singular two-parameter eigenvalue problems Andrej Muhi c - - PowerPoint PPT Presentation

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

On singular two-parameter eigenvalue problems

Andrej Muhiˇ c

Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Slovenia

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, May 10-17, 2009 This is joint work with B. Plestenjak.

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Outline

1

Two parameter eigenvalue problem

2

From MEP to a coupled GEP

3

From common regular eigenvector to eigenvalue of MEP

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Two-parameter eigenvalue problem

Two-parameter eigenvalue problem: W1(λ, µ)x1 := (A1 + λB1 + µC1) x1 = 0 (MEP) W2(λ, µ)x2 := (A2 + λB2 + µC2) x2 = 0 Eigenvalue: a pair (λ, µ) satisfying (MEP) for nonzero x1 and x2. Eigenvector: the tensor product x1 ⊗ x2. Equivalent problem : finding common zeros of polynomials p1(λ, µ) = det(W1(λ, µ)) and p2(λ, µ) = det(W2(λ, µ)).

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Tensor product approach

∆i matrices on the space Cn1×n2 ∆0 = B1 ⊗ C2 − C1 ⊗ B2 ∆1 = C1 ⊗ A2 − A1 ⊗ C2 ∆2 = A1 ⊗ B2 − B1 ⊗ A2. MEP is nonsingular ⇐ ⇒ some combination of ∆i (usually ∆0) is nonsingular. MEP is eiquivalent to a coupled pair of generalized eigenvalue problems ∆1z = λ∆0z (GEP) ∆2z = µ∆0z, where z = x1 ⊗ x2. ∆−1

0 ∆1 and ∆−1 0 ∆2 commute.

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

The singular two-parameter eigenvalue problem

every combination a∆0 + b∆1 + c∆2 is singular pencils λ∆0 − ∆1 and µ∆0 − ∆2 are singular Model updating (Cottin 2001, Cottin and Reetz 2006): finite element models of multibody systems are updated to match the measured input-output data. Spectrum of delay-differential equations (Jahrlebring 2008) Linearization of QMEP (Muhiˇ c, Plestenjak 2008) the numerical method for the solution of the singular MEP

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Some definitions

eigenvalue ω is a finite regular eigenvalue of matrix pencil λB − A if and only if rank(ωB − A) < max

s∈C rank(sB − A) = nr.

Normal rank of Wi(λ, µ) : nrank(Wi(λ, µ)) = max

λ,µ∈C rank(Wi(λ, µ)), i = 1, 2.

A pair (λf , µf ) ∈ C2 is a finite regular eigevalue: rank(Wi(λf , µf )) < nrank(Wi(λf , µf )), i = 1, 2. Geometric multiplicity of (λf , µf ) is

2

• i=1
• rank(Wi(λf , µf )) − nrank(Wi(λf , µf )
• .

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Our assumptions

The two-parameter eigenvalue problem is regular, pencils W1(λ, µ) and W2(λ, µ) have full normal rank, nrank(Wi(λ, µ)) = ni. p1(λ, µ) = det(W1(λ, µ)) ≡ 0, p2(λ, µ) = det(W2(λ, µ)) ≡ 0 finitely many eigenvalues ⇒ no common factor of p1 and p2 All factors of polynomials pi(λ, µ) = det(Wi(λ, µ)), i = 1, 2 depend

• n both variables λ and µ.

(λ2 + 1)(µ + 2) (λ3 + λ2µ + µ2 + 1)

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Eigenvalues of GEP

If MEP is nonsingular, eigenvalues coincide with eigenvalues of GEP. Definition A pair (λ0, µ0) ∈ C2 is a finite regular eigenvalue of matrix pencils ∆1 − λ∆0 and ∆2 − µ∆0 if the following is true: a) λ0 is a finite regular eigenvalue of ∆1 − λ∆0, b) µ0 is a finite regular eigenvalue of ∆2 − µ∆0, c) there exists a common regular eigenvector z, i.e., z = 0 such that (∆1 − λ0∆0)z = 0, (∆2 − µ0∆0)z = 0, and z ∈ R(∆i, ∆0) for i = 1, 2. The geometric multiplicity of (λ0, µ0) is dim(N) − dim(N ∩ (R(∆1, ∆0) ∪ R(∆2, ∆0))), N = ker(∆1 − λ0∆0) ∩ ker(∆2 − µ0∆0).

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Nonlinear two-parameter eigenvalue problem

T1(λ, µ)x1 = (NEP) T2(λ, µ)x2 = 0, matrix Ti(., .) : C × C → Cni×ni is differentiable, i = 1, 2. x1 and x2 solutions for the eigenvalue (λf , µf ), x1 ⊗ x2 corresponding right eigenvector. left eigenvector y1 ⊗ y2, where y∗

i Ti(λf , µf ) = 0, i = 1, 2.

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Theorem Let (λf , µf ) be an algebraically and geometrically simple eigenvalue of the nonlinear two-parameter eigenvalue problem (NEP) and x1 ⊗ x2, y1 ⊗ y2 corresponding right and left eigenvector such that x12 = x22 = y12 = y22 = 1. Then the matrix M0 :=

• y∗

1 ∂T1 ∂λ (λf , µf )x1

y∗

1 ∂T1 ∂µ (λf , µf )x1

y∗

2 ∂T2 ∂λ (λf , µf )x2

y∗

2 ∂T2 ∂µ (λf , µf )x2

• is nonsingular.

Connection between the Jacobian matrix of the polynomial system and the matrix

• y∗

1B1x1

y∗

1C1x1

y∗

2B2x2

y∗

2C2x2

• for the nonsingular right definite two-parameter eigenvalue problem

(Hochstenbach, Plestenjak, 2003)

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Sketch of the proof

define nonsingular Si(λ, µ) = Ti(λ, µ) yi x∗

i

• define αi(λ, µ) = eT

ni+1Si(λ, µ)−1eni+1, qi(λ, µ) = det(Si(λ, µ)).

ri(λ, µ) = det(Ti(λ, µ)) = αi(λ, µ)qi(λ, µ), ri(λf , µf ) = 0 and qi(λf , µf ) = 0. prove that

• ∂r1

∂λ (λf , µf ) ∂r1 ∂µ (λf , µf ) ∂r2 ∂λ (λf , µf ) ∂r2 ∂µ (λf , µf )

• = −

q1(λf , µf ) q2(λf , µf )

• M0.

Eigenvalue (λf , µf ) is simple ⇒ Jacobian is nonsingular ⇒ the matrix M0 is nonsingular

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

∆0 product

(λ∗, µ∗) an algebraically simple eigenvalue of MEP x1 ⊗ x2 and y1 ⊗ y2 corresponding right and left eigenvector (y1 ⊗ y2)∗∆0(x1 ⊗ x2) =

• y∗

1B1x1

y∗

1C1x1

y∗

2B2x2

y∗

2C2x2

• = 0

a generalization of the Rayleigh quotient a known result for a right definite MEP, ∆0 positive definitive a generalization of Koˇ sir’s result for the nonsingular case

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Characterization of a finite regular eigenvalue

Collorary Let λf be an eigenvalue of the matrix pencil A + λB with the corresponding right and left eigenvector x and y, respectively. If y∗Bx = 0 then λf is a finite regular eigenvalue. u(λf ) = x (polynomial solution equal to x at λf ) differentiate (A + λB)u(λ) = 0. Bu(λ) + (A + λB)u′(λ) = 0 ⇒ y∗Bu(λf ) + y∗(A + λf B)

• u′(λf ) = y∗Bx = 0

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

From MEP to coupled GEP

Theorem Every algebraically simple eigenvalue (λ0, µ0) of a regular two-parameter eigenvalue problem MEP is a finite regular eigenvalue of the associated pair

• f generalized eigenvalue problems GEP.

an algebraically simple eigenvalue (λ0, µ0) of MEP (y1 ⊗ y2)∗∆0x1 ⊗ x2 = 0 a common regular eigenvalue of the associated pair GEP a common regular eigenvalue of GEP →? an eigenvalue of MEP there is a bidirectonal connection between MEP and GEP

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Kronecker chains of A − λB

finite block Jp(α) (A − αB)u1 = 0, (A − αB)ui+1 = Bui, i = 1, . . . , p − 1. infinite block Np(α) Bu1 = 0, Bui+1 = Aui, i = 1, . . . , p − 1. right singular block Lp(α) Au1 = 0, Aui+1 = Bui, i = 1, . . . , p, = Bup+1. left singular block LT

p(α)

Aui = Bui+1, i = 1, . . . , p − 1.

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Kernel of A ⊗ D − B ⊗ C (Koˇ sir)

pencils A − λB and C − µD are regular a basis for the kernel of ∆ = A ⊗ D − B ⊗ C? a pencil pair (A − λB,C − µD). a) Jp1(α1) and Jp2(α2), where α1 = α2 = α. b) Np1 and Np2, Pair of Kronecker chains a) (similarly for b)) (A − αB)u1 = 0, (A − αB)ui+1 = Bui, i = 1, . . . , p1 − 1 (C − αD)v1 = 0, (C − αD)vi+1 = Bvi, i = 1, . . . , p2 − 1. Corresponding vectors in the kernel: zj =

j

• i=1

ui ⊗ vj+1−i, j = 1, . . . , min(p1, p2). Union of all vectors corresponding to pairs at a) or b) is a basis for ∆.

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Kernel of λ∆0 − ∆1

Vectors in the kernel of λ0∆0 − ∆1 = W1(λ0, 0) ⊗ C2 − C1 ⊗ W2(λ0, 0) are associated with the

pairs of Kronecker blocks (Np1, Np2), (Jp1, Jp2) of regular matrix pencils W1(λ0, 0) − α1C1 and W2(λ0, 0) − α2C2 zj =

j

• i=1

ui ⊗ vj+1−i, j = 1, . . . , p = min{p1, p2} pair (Np1, Np2) ∆0zj = −∆2zj−1, j = 2, ..., p ∆0z1 = ∆2z1 = 0 m pairs of blocks (Np1, Np2), associated vectors zk1, . . . , zkmk, k = 1, . . . , m no pair of blocks (Jp1, Jp2), a potential common regular eigenvector z =

m

• j=1

mj

• k=1

ξjkzjk is in the singular part of µ∆0 − ∆2

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

a) If the pencil ∆1 − λ∆0 is singular, then it contains at least one block L0. b)

λ0 is not a regular eigenvalue of the pencil ∆1 − λ∆0 Wi(λ0, 0) − αiCi is nonsingular, i = 1, 2 Kronecker canonical form of Wi(λ0, 0) − αiCi contains infinite regular blocks Np1, . . . , Npki for i = 1, 2

rank(∆1 − λ0∆0) = N −

k1

• i=1

k2

• j=1

min(pi, pj) The Kronecker canonical form of the pencil ∆1 − λ∆0 contains at least k1k2 blocks L0.

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Common regular part

common regular eigenvector ⇒ pair of blocks Jp1(α), Jp2(α) algebraically simple eigenvalues of MEP m1 geometric multiplicity of λf , independent eigenvectors x11 ⊗ x21, . . . , x1m1 ⊗ x2m1, (λf , −αi) eigenvalues of GEP m2 geometric multiplicity of µf , independent eigenvectors u11 ⊗ u21, . . . , u1m2 ⊗ u2m2, (−βj, µf ) eigenvalues of GEP Can λf = βj hold for j = 1, . . . , m1?

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Bidirectional link

Theorem Eigenvalue (λ0, µ0) of GEP is a finite regular eigenvalue of the regular two-parameter eigenvalue problem MEP. Theorem Every algebraically simple eigenvalue (λ0, µ0) of a regular two-parameter eigenvalue problem MEP is a finite regular eigenvalue of the associated pair

• f generalized eigenvalue problems GEP.

Assumptions: all finite eigenvalues of regular MEP are algebraically simple polynomials p1 and p2 do not have a factor that depends on exactly one

• f the variables λ or µ

λ0 and µ0 are nondefective eigenvalues of ∆1 − λ∆0 and ∆2 − µ∆0

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Conclusions

A bidirectional connection between singular MEP and GEP. Solution for regular singular MEP in the generic case. Eigenvalues can be computed using algorithm presented in QMEP. Work in progress: the singular MEP A bidirectional link in the nongeneric case? Regular singular MEP and a common factor ? More than two parameters? . . .

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Example 1

W1(λ, µ) = (A1 + λB1 + µC1) =

• −λ − µ

−1 −1 1

• ,

W2(λ, µ) = (A2 + λB2 + µC2) = −2λ + µ −1 −1 2

• p1(λ, µ)

= det(W1(λ, µ)) = −λ − µ + 1 p2(λ, µ) = det(W2(λ, µ)) = −4λ + 2µ + 1 regular problem with eigenvalue (λ, µ) = ( 1

2, 1 2).

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

∆0 =    −3    , ∆1 =    −1 −1 −1 2 −1 1    , ∆2 =    1 −2 1 −2 −2 2   

Kronecker structure of ∆1 − λ∆0 and ∆2 − µ∆0 is L0, 2N1, LT

0, and

J1( 1

2)

minimal reducing subspace R(∆1, ∆0) = R(∆2, ∆0) = span(e4) subspace for the two blocks N1 is span(e2, e3) a common regular eigenvector 2 1 2 1T + αe4

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Example 2

W1(λ, µ) =   2 + λ 1 + 2λ λ λ 2 + 2λ + 2µ µ µ 1 + 2µ 2 + µ   , W2(λ, µ) =   1 + λ 1 + 2λ λ λ 1 + 2λ + 2µ µ µ 1 + 2µ 1 + µ   . p1(λ, µ) = λ2 + 6µλ + 10λ + µ2 + 10µ + 8 p2(λ, µ) = (λ + µ + 1)2 a quadruple eigenvalue (λ, µ) = (− 1

2, − 1 2) that is geometrically simple

The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector

Kronecker structure L0, 2N2, LT

0, J4(− 1 2)

a common regular right eigenvector z1 = 1 0T ⊗ 1 −1 1T , a common regular left eigenvector y =

• 1

2 1

• ⊗
• 1

1 T y∗∆0z1 = 0 condition of the theorem is not satisfied algorithm works anyway!