Bounds for the spectrum of a two parameter matrix eigenvalue problem

Michael Gil'

doi:10.1016/j.laa.2015.05.008

Bounds for the spectrum of a two parameter matrix eigenvalue problem

Michael Gil'

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

We consider the two parameter eigenvalue problem T_jv_j - λ₁A_j1v_j - λ₂A_j2v_j = 0, where λ_j C; T_j, A_jk (j,k=1,2) are matrices. Bounds for the spectral radius of that problem are suggested. Our main tool is a norm estimate for the operator inverse to the operator A₁₁ ⊗ A₂₂ - A₁₂ ⊗ A₂₁, where ⊗ means the tensor product. In addition, by virtue of that norm estimate and the Ostrowsky-Schneider theorem we establish a condition that provides the conservation of the number of the eigenvalues of the considered problem in a half-plane.

Original language	English
Pages (from-to)	201-218
Number of pages	18
Journal	Linear Algebra and Its Applications
Volume	498
DOIs	https://doi.org/10.1016/j.laa.2015.05.008
State	Published - 1 Jun 2016

Keywords

Ostrowsky-Schneider theorem
Spectrum
Two parameter matrix eigenvalue problem

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1016/j.laa.2015.05.008

Cite this

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title = "Bounds for the spectrum of a two parameter matrix eigenvalue problem",

abstract = "We consider the two parameter eigenvalue problem Tjvj - λ1Aj1vj - λ2Aj2vj = 0, where λj C; Tj, Ajk (j,k=1,2) are matrices. Bounds for the spectral radius of that problem are suggested. Our main tool is a norm estimate for the operator inverse to the operator A11 ⊗ A22 - A12 ⊗ A21, where ⊗ means the tensor product. In addition, by virtue of that norm estimate and the Ostrowsky-Schneider theorem we establish a condition that provides the conservation of the number of the eigenvalues of the considered problem in a half-plane.",

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AU - Gil', Michael

PY - 2016/6/1

Y1 - 2016/6/1

N2 - We consider the two parameter eigenvalue problem Tjvj - λ1Aj1vj - λ2Aj2vj = 0, where λj C; Tj, Ajk (j,k=1,2) are matrices. Bounds for the spectral radius of that problem are suggested. Our main tool is a norm estimate for the operator inverse to the operator A11 ⊗ A22 - A12 ⊗ A21, where ⊗ means the tensor product. In addition, by virtue of that norm estimate and the Ostrowsky-Schneider theorem we establish a condition that provides the conservation of the number of the eigenvalues of the considered problem in a half-plane.

AB - We consider the two parameter eigenvalue problem Tjvj - λ1Aj1vj - λ2Aj2vj = 0, where λj C; Tj, Ajk (j,k=1,2) are matrices. Bounds for the spectral radius of that problem are suggested. Our main tool is a norm estimate for the operator inverse to the operator A11 ⊗ A22 - A12 ⊗ A21, where ⊗ means the tensor product. In addition, by virtue of that norm estimate and the Ostrowsky-Schneider theorem we establish a condition that provides the conservation of the number of the eigenvalues of the considered problem in a half-plane.

KW - Ostrowsky-Schneider theorem

KW - Spectrum

KW - Two parameter matrix eigenvalue problem

UR - https://www.scopus.com/pages/publications/84929591359

U2 - 10.1016/j.laa.2015.05.008

DO - 10.1016/j.laa.2015.05.008

M3 - Article

AN - SCOPUS:84929591359

SN - 0024-3795

VL - 498

SP - 201

EP - 218

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

Bounds for the spectrum of a two parameter matrix eigenvalue problem

Abstract

Keywords

ASJC Scopus subject areas

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