Low rank perturbations of strongly definitizable transformations and matrix polynomials

Peter Lancaster; Alexander S. Markus; Qiang Ye

doi:10.1016/0024-3795(94)90479-0

Low rank perturbations of strongly definitizable transformations and matrix polynomials

Peter Lancaster, Alexander S. Markus, Qiang Ye

Mathematics

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7 Scopus citations

Abstract

Let A₀ be a transformation on a finite dimensional Hilbert space which is self-adjoint in an indefinite scalar product generated by G₀ (G^*₀ and invertible). The spectrum of A₀ is real when A₀ is G₀-strongly definitizable. The problems considered here concern the number of real eigenvalues of a G-self-adjoint transformation A where A and G are low rank perturbations of A₀ and G₀. A notion called the "order of neutrality" of A with respect to G is introduced which is relevant to this problem area. Using linearization as well as direct methods, results are obtained concerning self-adjoint matrix polynomials which are low rank perturbationsof (suitably defined) definitizable matrix polynomials. Applications are made to quadratic matrix polynomials arising in the study of damped systems and gyroscopic systems.

Original language	English
Pages (from-to)	3-29
Number of pages	27
Journal	Linear Algebra and Its Applications
Volume	197-198
Issue number	C
DOIs	https://doi.org/10.1016/0024-3795(94)90479-0
State	Published - 1994

Bibliographical note

Funding Information:
*The work of these authors was supported in part by grants from the Natural Sciences and

Funding Information:
Engineering Research Council of Canada and the Israel Ministry of Science, respectively.

ASJC Scopus subject areas

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

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title = "Low rank perturbations of strongly definitizable transformations and matrix polynomials",

abstract = "Let A0 be a transformation on a finite dimensional Hilbert space which is self-adjoint in an indefinite scalar product generated by G0 (G*0 and invertible). The spectrum of A0 is real when A0 is G0-strongly definitizable. The problems considered here concern the number of real eigenvalues of a G-self-adjoint transformation A where A and G are low rank perturbations of A0 and G0. A notion called the {"}order of neutrality{"} of A with respect to G is introduced which is relevant to this problem area. Using linearization as well as direct methods, results are obtained concerning self-adjoint matrix polynomials which are low rank perturbationsof (suitably defined) definitizable matrix polynomials. Applications are made to quadratic matrix polynomials arising in the study of damped systems and gyroscopic systems.",

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T1 - Low rank perturbations of strongly definitizable transformations and matrix polynomials

AU - Lancaster, Peter

AU - Markus, Alexander S.

AU - Ye, Qiang

N1 - Funding Information: *The work of these authors was supported in part by grants from the Natural Sciences and Funding Information: Engineering Research Council of Canada and the Israel Ministry of Science, respectively.

PY - 1994

Y1 - 1994

N2 - Let A0 be a transformation on a finite dimensional Hilbert space which is self-adjoint in an indefinite scalar product generated by G0 (G*0 and invertible). The spectrum of A0 is real when A0 is G0-strongly definitizable. The problems considered here concern the number of real eigenvalues of a G-self-adjoint transformation A where A and G are low rank perturbations of A0 and G0. A notion called the "order of neutrality" of A with respect to G is introduced which is relevant to this problem area. Using linearization as well as direct methods, results are obtained concerning self-adjoint matrix polynomials which are low rank perturbationsof (suitably defined) definitizable matrix polynomials. Applications are made to quadratic matrix polynomials arising in the study of damped systems and gyroscopic systems.

AB - Let A0 be a transformation on a finite dimensional Hilbert space which is self-adjoint in an indefinite scalar product generated by G0 (G*0 and invertible). The spectrum of A0 is real when A0 is G0-strongly definitizable. The problems considered here concern the number of real eigenvalues of a G-self-adjoint transformation A where A and G are low rank perturbations of A0 and G0. A notion called the "order of neutrality" of A with respect to G is introduced which is relevant to this problem area. Using linearization as well as direct methods, results are obtained concerning self-adjoint matrix polynomials which are low rank perturbationsof (suitably defined) definitizable matrix polynomials. Applications are made to quadratic matrix polynomials arising in the study of damped systems and gyroscopic systems.

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Low rank perturbations of strongly definitizable transformations and matrix polynomials

Abstract

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