[PPT] - Indefinite linear matrix pencils and the multi-eigenvalue problem PowerPoint Presentation

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Indefinite linear matrix pencils and the multi-eigenvalue problem

Hasen Mekki ÖZTÜRK

University of Reading

Based on joint work with Michael Levitin August, 2017 IWOTA

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Outline

1

Pencils

2

A matrix pencil example

3

Multi-Parametric Eigenvalue Problem

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Pencils

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Pencils

Self-adjoint pencils

Let P = P (λ) := T0 + λT1 + λ2T2 + . . . + λmTm, be a family of (bounded) operators in a Hilbert space H, which depends on a spectral parameter λ ∈ C, with self-adjoint operator coefficients Tj = (Tj)∗ , j = 1, . . . , n. Such a family is called a self-adjoint (polynomial) operator pencil. I shall only deal with a linear self-adjoint operator pencil written as P (λ) = T − λS.

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Pencils

Spectrum of a linear pencil

λ0 is an eigenvalue of P if there exists x ∈ H {0} such that P (λ0) x = 0, i.e., if 0 is an eigenvalue of P (λ0). The spectrum is the set of values λ0 for which there is no bounded inverse P (λ0)−1, i.e. if 0 ∈ SpecP (λ0). For a linear pencil, the eigenvalue problem becomes (T − λS) x = 0, and if S is invertible, then it reduces to the eigenvalue problem for a (non-self-adjoint) operator S−1T; S−1Tx = λx.

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Pencils

If either T or S is sign-definite, then the problem may be reduced to the one for a self-adjoint operator S−1/2TS−1/2, and the spectrum is real. If, however, both T, S are sign-indefinite, then the spectrum may be non-real.

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A matrix pencil example

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A matrix pencil example

Example - A matrix pencil [DaLe]

Fix an integer n ∈ N, N = 2n, and define the N × N classes of matrices H(N)

c

and S, where H(N)

c

:=       c 1 1 c ... ... ... 1 1 c       , S := In −In

where c ∈ R is a parameter and In is the identity matrix.

The behavior of eigenvalues of the linear operator pencil P := P (λ) = H(N)

c

− λS as N → ∞ was studied by Davies & Levitin(2014).

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A matrix pencil example

Example - A matrix pencil [DaLe]

If |c| ≥ 2, then Spec (P) ⊂ R. Spec (P) is invariant under the symmetry c → −c. Spec (P) is symmetric with respect to Re λ = 0 and Im λ = 0. Davies & Levitin studied the asymptotic behaviour of eigenvalues of P for large n;

For c = 0, |Im λ| ∼ 1 nY0 (|Re λ|) . For 0 < c < 2, |Im λ| 1 nYc (|Re λ|) .

Functions Y0 and Yc are explicit (though rather complicated), and have logarithmic singularities at Re λ = 0.

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A matrix pencil example

Example - A matrix pencil [DaLe]

Video time!

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A matrix pencil example

Spec (P) for c = √ 5/2 and n = 500, 250, 99.

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A matrix pencil example

∪250

m=100Spec (P) for c =

√ 5/2

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A matrix pencil example

Conjecture

Asymptotic and numerical evidence suggest the following: Let c > 0. If λ ∈ Spec (P (λ)) R, then c < 2 and |λ ± c| < 2. This can also be translated in terms of Chebyshev polynomials via explicit expression for det

H(N)

c

− λS

:

Let σ, τ ∈ C, Im (σ) = Im (τ) > 0. If, for some n ∈ N, Un+1 (σ) Un+1 (τ) + Un (σ) Un (τ) = 0, then |σ| < 1 and |τ| < 1.

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Multi-Parametric Eigenvalue Problem

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Multi-Parametric Eigenvalue Problem

Pencil to Parametric problem

Recall the pencil P; H(N)

c

− λS =            c − λ 1 1 ... ... ... c − λ 1 1 c + λ ... ... ... 1 1 c + λ            . Denote α = λ − c, β = −λ − c.

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Multi-Parametric Eigenvalue Problem

Pencil to Parametric problem

We will act by P on vectors which we will write as (u1, . . . , un, vn, . . . , v1)T . Then

H(n)

− αIn B B H(n) − βIn − → u − → v

= −

→ 0 , where B = B∗ with Bnn = 1 and all other entries of B are zeros.

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Multi-Parametric Eigenvalue Problem

Pencil to Parametric problem

We first generalize to the following: for any κ > 0, let B = κP, P = P∗, P = 1,

H(n)

− αIn κP κP H(n) − βIn − → u − → v

= −

→ 0 . (1) which is a special case of A − αI1 C C ∗ D − βI2 − → u − → v

= −

→ 0 , (2) where, in general, − → u ∈ H1 and − → v ∈ H2, A, D are self-adjoint operators in H1, H2, respectively, C is a linear operator from H2 to H1, α, β ∈ C are spectral parameters.

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Multi-Parametric Eigenvalue Problem

Two-parameter Matrix Eigenvalue problem

Denote M = A C C ∗ D

,

so that the problem

M −

αI1 βI2 − → u − → v

= −

→ 0 , (3) where − → u ∈ H1 and − → v ∈ H2. (α, β) ∈ C2 a multi-eigenvalue (or a pair-eigenvalue) of M if there exists a non-trivial solution − → u − → v

∈ H of (3).

We denote by Specp (M) the spectrum of pair-eigenvalues of M. If α, β ∈ R, then (α, β) is called as a real pair-eigenvalue, and

therwise it is a non-real pair-eigenvalue of (3).

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Multi-Parametric Eigenvalue Problem

β (α) problem

The equation (3) can be re-written as

(A − αI1) −

→ u = −C− → v , (D − βI2) − → v = −C ∗− → u . If α / ∈ Spec (A) and β is an eigenvalue of

D − C ∗ (A − αI1)−1 C
,

then (α, β) ∈ Specp (M). Note: α and β are interchangeable.

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Multi-Parametric Eigenvalue Problem

Restrictions

Now, suppose that H1, H2 are finite dimensional, and therefore we are dealing with matrices. Additionally dim H1 = dim H2. C has rank 1, or C = κP, where κ > 0 and P is a projection on a

ne-dimensional subspace span {−

→ ϕ } of H,

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Multi-Parametric Eigenvalue Problem

Notations

The restriction of X on the space of vectors orthogonal to − → ϕ will be denoted by X⊥,⊥. Eigenvalues of A and D will be denoted by

α1

≥

α2 ≥ . . . ≥

αn,

β1

≥

β2 ≥ . . . ≥

βn, respectively. Eigenvalues of A⊥,⊥ and D⊥,⊥ will be denoted by

α1

≥

α2 ≥ . . . ≥

αn−1,

β1

≥

β2 ≥ . . . ≥

βn−1, respectively.

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Multi-Parametric Eigenvalue Problem

Remark

All numerical examples will be related to

H(n)

− αI κP κP H(n) − βI − → u − → v

= −

→ 0 . All theoretical results will be related to A − αI κP κP D − βI − → u − → v

= −

→ 0 .

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Multi-Parametric Eigenvalue Problem

Spectral picture for n = 7

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Multi-Parametric Eigenvalue Problem

Spectral picture for n = 7

Blue curves are all real eigencurves β (α), α ∈ R. Red curves are graphs of Reβ (Reα) for eigenpairs such that Im (α + β) = 0, which keeps all (α, β) ∈ R2 in the picture and some complex pair-eigenvalues. Lemma Blue and red lines intersect iff d dαβ (α) = −1.

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Multi-Parametric Eigenvalue Problem

Characteristic equation

Theorem If α / ∈ Spec (A) and β / ∈ Spec (D), then the characteristic equation for (α, β) ∈ Specp (M) is κ2 (A − αIn)−1 − → ϕ , − → ϕ (D − βIn)−1 − → ϕ , − → ϕ

= 1,

(4) which implies β

′ = −κ2

(A − αIn)−2 −

→ ϕ , − → ϕ (D − βIn)−1 − → ϕ , − → ϕ 2

(D − βIn)−2 −

→ ϕ , − → ϕ

,

and therefore dβ

dα < 0 on each real branch of β (α), α ∈ R.

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Multi-Parametric Eigenvalue Problem

Mesh

R1,3 R2,1 R1,2 R1,4 R3,1 R2,2 R3,2 R2,3 R4,1 R1,1 Α

1

Α

2

Β

1

Β

2

Α

1

Α

2

Β

1

Β

2

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Multi-Parametric Eigenvalue Problem

Spectral picture for n = 7

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Multi-Parametric Eigenvalue Problem

n = 4 and particular values of κ

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Multi-Parametric Eigenvalue Problem

Chess Board Structure for n = 6

Figure : Superimposing the values of κ from 0.001 to 10 with the step-size of 0.1.

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Multi-Parametric Eigenvalue Problem

Chess Board Structure

Chess Board Theorem: Suppose that Spec (A) Spec (A⊥,⊥) = ∅ and Spec (D) Spec (D⊥,⊥) = ∅. Then all real pair-eigenvalues (α, β) of M lies in the region Rp,q where p + q is even, i.e. (α, β) ⊂ R2 ⇒ (α, β) ∈ Rp,q.

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Multi-Parametric Eigenvalue Problem

When α ∈ Spec (A)

Lemma Suppose α = αi ∈ Spec (A), i = 1, . . . , n, and let − → ψi be an eigenfunction corresponding to the eigenvalue αi of A. Assume that − → ϕ , − → ψi

= 0.

Then, for any κ ∈ R {0}, (α, β (α)) ∈ Specp (M)

β (α)

∈ Spec (D⊥⊥) , and additionally for α ≈ αi, there exists one (α, β (α)) ∈ Specp (M) such that β (α) → ±∞ as α → α±

i .

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Multi-Parametric Eigenvalue Problem

As κ → 0

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Multi-Parametric Eigenvalue Problem

As κ → 0

Theorem Let κ → 0. Then for every i ∈ {1, . . . , n} and every β ∈ R, there exists a sequence

(αk, βk) ∈ Specp (M)
k such that

αk → αi, βk → β, similarly for every i ∈ {1, . . . , n} and every α ∈ R, there exists a sequence

(αk, βk) ∈ Specp (M)
k such that

αk → α, βk → βi.

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Multi-Parametric Eigenvalue Problem

As κ → +∞

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Multi-Parametric Eigenvalue Problem

As κ → +∞

Theorem Let κ → ∞. Then for every j ∈ {1, . . . , n − 1} and every β ∈ R, there exists a sequence

(αk, βk) ∈ Specp (M)
k such that

αk → αj, βk → β, similarly for every j ∈ {1, . . . , n − 1} and every α ∈ R, there exists a sequence

(αk, βk) ∈ Specp (M)
k such that

αk → α, βk → βj. In addition, there exists one non-real family of pair-eigenvalue (α, β) of M such that α = β.

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Multi-Parametric Eigenvalue Problem

Modified problem: Non-real collisions

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Multi-Parametric Eigenvalue Problem

E. Brian Davies and Michael Levitin, Spectra of a class of

non-self-adjoint matrices, Linear Algebra and its Applications 448, 55-84 (2014).

C. Tretter, Spectral Theory of Block Operator Matrices and

Applications, Imperial Collefe Press, London, UK, 2008. Atkinson, F. V., 1968. Multiparameter spectral theory. Bulletin of the American Mathematical Society, 74(1), 1-27. Atkinson, F. V., 1972. Multiparameters eigenvalue problems. Academic Press. E.B. Davies, Linear Operators and their Spectra, Cambridge University Press, Cambridge, UK, 2007.

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