A variational principle for eigenvalues of pencils of Hermitian matrices

Paul Binding; Branko Najman; Qiang Ye

doi:10.1007/BF01228041

A variational principle for eigenvalues of pencils of Hermitian matrices

Paul Binding, Branko Najman, Qiang Ye

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

13 Scopus citations

Abstract

Let H = (A, B) be a pair of Hermitian N × N matrices. A complex number λ is an eigenvalue of H if det(A - λB) = 0 (we include λ = ∞ if detB = 0). For nonsingular H (i.e., for which some λ is not an eigenvalue), we show precisely which eigenvalues can be characterized as σ⁺_k = sup{inf{x* Ax : x* Bx = 1, x ∈ S}, S ∈ S_k}, S_k being the set of subspaces of C^N of codimension k - 1.

Original language	English
Pages (from-to)	398-422
Number of pages	25
Journal	Integral Equations and Operator Theory
Volume	35
Issue number	4
DOIs	https://doi.org/10.1007/BF01228041
State	Published - Nov 1999

Bibliographical note

Funding Information:
the Ministry of Science of Croatia *Research supported by NSERC of Canada

Funding Information:
*Research supported by NSERC of Canada and the l.W.Killam Foundation tProfessor Najman died suddenly while this work was at its final stage. His research was supported by

Funding

the Ministry of Science of Croatia *Research supported by NSERC of Canada *Research supported by NSERC of Canada and the l.W.Killam Foundation tProfessor Najman died suddenly while this work was at its final stage. His research was supported by

Funders	Funder number
NSERC of Canada
Natural Sciences and Engineering Research Council of Canada
Ministry of Science and Technology, Croatia

ASJC Scopus subject areas

Analysis
Algebra and Number Theory

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10.1007/BF01228041

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title = "A variational principle for eigenvalues of pencils of Hermitian matrices",

abstract = "Let H = (A, B) be a pair of Hermitian N × N matrices. A complex number λ is an eigenvalue of H if det(A - λB) = 0 (we include λ = ∞ if detB = 0). For nonsingular H (i.e., for which some λ is not an eigenvalue), we show precisely which eigenvalues can be characterized as σ+k = sup{inf{x* Ax : x* Bx = 1, x ∈ S}, S ∈ Sk}, Sk being the set of subspaces of CN of codimension k - 1.",

author = "Paul Binding and Branko Najman and Qiang Ye",

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AU - Najman, Branko

AU - Ye, Qiang

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PY - 1999/11

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N2 - Let H = (A, B) be a pair of Hermitian N × N matrices. A complex number λ is an eigenvalue of H if det(A - λB) = 0 (we include λ = ∞ if detB = 0). For nonsingular H (i.e., for which some λ is not an eigenvalue), we show precisely which eigenvalues can be characterized as σ+k = sup{inf{x* Ax : x* Bx = 1, x ∈ S}, S ∈ Sk}, Sk being the set of subspaces of CN of codimension k - 1.

AB - Let H = (A, B) be a pair of Hermitian N × N matrices. A complex number λ is an eigenvalue of H if det(A - λB) = 0 (we include λ = ∞ if detB = 0). For nonsingular H (i.e., for which some λ is not an eigenvalue), we show precisely which eigenvalues can be characterized as σ+k = sup{inf{x* Ax : x* Bx = 1, x ∈ S}, S ∈ Sk}, Sk being the set of subspaces of CN of codimension k - 1.

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IS - 4

ER -

A variational principle for eigenvalues of pencils of Hermitian matrices

Abstract

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