Bounds for eigenvalues of matrix polynomials

Nicholas J. Higham; Françoise Tisseur

doi:10.1016/S0024-3795(01)00316-0

Bounds for eigenvalues of matrix polynomials

Nicholas J. Higham
, Françoise Tisseur

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

Abstract

Upper and lower bounds are derived for the absolute values of the eigenvalues of a matrix polynomial (or λ-matrix). The bounds are based on norms of the coefficient matrices and involve the inverses of the leading and trailing coefficient matrices. They generalize various existing bounds for scalar polynomials and single matrices. A variety of tools are used in the derivations, including block companion matrices, Gershgorin's theorem, the numerical radius, and associated scalar polynomials. Numerical experiments show that the bounds can be surprisingly sharp on practical problems. © 2002 Elsevier Science Inc.

Original language	English
Pages (from-to)	5-22
Number of pages	17
Journal	Linear Algebra and its Applications
Volume	358
Issue number	1-3
DOIs	https://doi.org/10.1016/S0024-3795(01)00316-0
Publication status	Published - 1 Jan 2003

Keywords

λ-Matrix
Block companion matrix
Gershgorin's theorem
Matrix polynomial
Numerical radius
Polynomial eigenvalue problem

Access to Document

10.1016/S0024-3795(01)00316-0

Cite this

@article{f556a5a7d4444dc0a59d24e1e8b111f3,

title = "Bounds for eigenvalues of matrix polynomials",

abstract = "Upper and lower bounds are derived for the absolute values of the eigenvalues of a matrix polynomial (or λ-matrix). The bounds are based on norms of the coefficient matrices and involve the inverses of the leading and trailing coefficient matrices. They generalize various existing bounds for scalar polynomials and single matrices. A variety of tools are used in the derivations, including block companion matrices, Gershgorin's theorem, the numerical radius, and associated scalar polynomials. Numerical experiments show that the bounds can be surprisingly sharp on practical problems. {\textcopyright} 2002 Elsevier Science Inc.",

keywords = "λ-Matrix, Block companion matrix, Gershgorin's theorem, Matrix polynomial, Numerical radius, Polynomial eigenvalue problem",

author = "Higham, \{Nicholas J.\} and Fran{\c c}oise Tisseur",

year = "2003",

month = jan,

day = "1",

doi = "10.1016/S0024-3795(01)00316-0",

language = "English",

volume = "358",

pages = "5--22",

journal = "Linear Algebra and its Applications",

issn = "0024-3795",

publisher = "Elsevier BV",

number = "1-3",

}

TY - JOUR

T1 - Bounds for eigenvalues of matrix polynomials

AU - Higham, Nicholas J.

AU - Tisseur, Françoise

PY - 2003/1/1

Y1 - 2003/1/1

N2 - Upper and lower bounds are derived for the absolute values of the eigenvalues of a matrix polynomial (or λ-matrix). The bounds are based on norms of the coefficient matrices and involve the inverses of the leading and trailing coefficient matrices. They generalize various existing bounds for scalar polynomials and single matrices. A variety of tools are used in the derivations, including block companion matrices, Gershgorin's theorem, the numerical radius, and associated scalar polynomials. Numerical experiments show that the bounds can be surprisingly sharp on practical problems. © 2002 Elsevier Science Inc.

AB - Upper and lower bounds are derived for the absolute values of the eigenvalues of a matrix polynomial (or λ-matrix). The bounds are based on norms of the coefficient matrices and involve the inverses of the leading and trailing coefficient matrices. They generalize various existing bounds for scalar polynomials and single matrices. A variety of tools are used in the derivations, including block companion matrices, Gershgorin's theorem, the numerical radius, and associated scalar polynomials. Numerical experiments show that the bounds can be surprisingly sharp on practical problems. © 2002 Elsevier Science Inc.

KW - λ-Matrix

KW - Block companion matrix

KW - Gershgorin's theorem

KW - Matrix polynomial

KW - Numerical radius

KW - Polynomial eigenvalue problem

U2 - 10.1016/S0024-3795(01)00316-0

DO - 10.1016/S0024-3795(01)00316-0

M3 - Article

SN - 0024-3795

VL - 358

SP - 5

EP - 22

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

IS - 1-3

ER -

Bounds for eigenvalues of matrix polynomials

Abstract

Keywords

Access to Document

Fingerprint

Cite this