Convexity and lipschitz behavior of small pseudospectra

J. V. Burke; A. S. Lewis; M. L. Overton

doi:10.1137/050645841

Convexity and lipschitz behavior of small pseudospectra

J. V. Burke, A. S. Lewis, M. L. Overton

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

Abstract

The ε-pseudospectrum of a matrix A is the subset of the complex plane consisting of all eigenvalues of complex matrices within a distance ε of A, measured by the operator 2-norm. Given a nonderogatory matrix A₀, for small ε > 0, we show that the ε-pseudospectrum of any matrix A near A₀ consists of compact convex neighborhoods of the eigenvalues of A₀. Furthermore, the dependence of each of these neighborhoods on A is Lipschitz.

Original language	English (US)
Pages (from-to)	586-595
Number of pages	10
Journal	SIAM Journal on Matrix Analysis and Applications
Volume	29
Issue number	2
DOIs	https://doi.org/10.1137/050645841
State	Published - 2007

Keywords

Eigenvalue optimization
Lipschitz multi-function
Nonsmooth analysis
Pseudospectrum
Robust optimization

ASJC Scopus subject areas

Analysis

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10.1137/050645841

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AU - Lewis, A. S.

AU - Overton, M. L.

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N2 - The ε-pseudospectrum of a matrix A is the subset of the complex plane consisting of all eigenvalues of complex matrices within a distance ε of A, measured by the operator 2-norm. Given a nonderogatory matrix A0, for small ε > 0, we show that the ε-pseudospectrum of any matrix A near A0 consists of compact convex neighborhoods of the eigenvalues of A0. Furthermore, the dependence of each of these neighborhoods on A is Lipschitz.

AB - The ε-pseudospectrum of a matrix A is the subset of the complex plane consisting of all eigenvalues of complex matrices within a distance ε of A, measured by the operator 2-norm. Given a nonderogatory matrix A0, for small ε > 0, we show that the ε-pseudospectrum of any matrix A near A0 consists of compact convex neighborhoods of the eigenvalues of A0. Furthermore, the dependence of each of these neighborhoods on A is Lipschitz.

KW - Eigenvalue optimization

KW - Lipschitz multi-function

KW - Nonsmooth analysis

KW - Pseudospectrum

KW - Robust optimization

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JF - SIAM Journal on Matrix Analysis and Applications

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Convexity and lipschitz behavior of small pseudospectra

Abstract

Keywords

ASJC Scopus subject areas

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