On the lengths of Jordan chains for eigenvalues on the boundary of numerical range of quadratic operator polynomial

Naum Krupnik

doi:10.1016/j.laa.2003.07.002

On the lengths of Jordan chains for eigenvalues on the boundary of numerical range of quadratic operator polynomial

Naum Krupnik

Department of Mathematics

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

1 Scopus citations

Abstract

We describe a connection between the geometrical structure of the numerical range W(L) of a selfadjoint quadratic operator polynomial L(λ) and the lengths of Jordan chains for eigenvalues on the boundary of the numerical range. Also for each pair of natural numbers m,n (1≤m≤2n) we construct a monic quadratic polynomial L_m,n(λ)=λ²+λC+B with selfadjoint n×n matrices C, B which has an eigenvalue λ₀∈∂W(L_m,n) with a Jordan chain of length m.

Original language	English
Pages (from-to)	131-147
Number of pages	17
Journal	Linear Algebra and Its Applications
Volume	385
Issue number	1-3
DOIs	https://doi.org/10.1016/j.laa.2003.07.002
State	Published - 1 Jul 2004

Keywords

Jordan chains of operator polynomial
Numerical range of operator
Numerical range of operator polynomial
Operator and matrix polynomials

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10.1016/j.laa.2003.07.002

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abstract = "We describe a connection between the geometrical structure of the numerical range W(L) of a selfadjoint quadratic operator polynomial L(λ) and the lengths of Jordan chains for eigenvalues on the boundary of the numerical range. Also for each pair of natural numbers m,n (1≤m≤2n) we construct a monic quadratic polynomial Lm,n(λ)=λ2+λC+B with selfadjoint n×n matrices C, B which has an eigenvalue λ0∈∂W(Lm,n) with a Jordan chain of length m.",

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AB - We describe a connection between the geometrical structure of the numerical range W(L) of a selfadjoint quadratic operator polynomial L(λ) and the lengths of Jordan chains for eigenvalues on the boundary of the numerical range. Also for each pair of natural numbers m,n (1≤m≤2n) we construct a monic quadratic polynomial Lm,n(λ)=λ2+λC+B with selfadjoint n×n matrices C, B which has an eigenvalue λ0∈∂W(Lm,n) with a Jordan chain of length m.

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KW - Operator and matrix polynomials

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On the lengths of Jordan chains for eigenvalues on the boundary of numerical range of quadratic operator polynomial

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