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2006.148: More on pseudospectra for polynomial eigenvalue problems and applications in control theory

2006.148: Nicholas J. Higham and Françoise Tisseur (2002) More on pseudospectra for polynomial eigenvalue problems and applications in control theory. Elsevier, Linear Algebra and its Applications, 351-352. pp. 435-453. ISSN 0024-3795

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DOI: 10.1016/S0024-3795(01)00542-0

Abstract

Definitions and characterizations of pseudospectra are given for rectangular matrix poly-nomials expressed in homogeneous form: P(α,β)=α^dA_d+α^{d−1}βA_{d−1}+...+β^dA_0. It is shown that problems with infinite (pseudo)eigenvalues are elegantly treated in this framework. For such problems stereographic projection onto the Riemann sphere is shown to provide a convenient way to visualize pseudospectra. Lower bounds for the distance to the nearest nonregular polynomial and the nearest uncontrollable dth order system (with equality for standard state-space systems) are obtained in terms of pseudospectra, showing that pseudospectra are a fundamental tool for reasoning about matrix polynomials in areas such as control theory. How and why to incorporate linear structure into pseudospectra is also discussed by example.

Item Type:Article
Uncontrolled Keywords:Polynomial eigenvalue problem; λ-Matrix; Matrix polynomial; Homogeneous form; Pseudospectrum; Stereographic projection; Riemann sphere; Nearest nonregular polynomial; Nearest uncontrollable system; Structured perturbations
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2006.148
Deposited By:Miss Louise Stait
Deposited On:27 June 2006

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