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Structured Pseudospectra and the condition of a nonderogatory eigenvalue

Please always quote using this URN:urn:nbn:de:0296-matheon-4192
  • Let $\lambda$ be a nonderogatory eigenvalue of $A \in \C^{n \times n}$. The sensitivity of $\lambda$ with respect to matrix perturbations $A \leadsto A+\Delta,\Delta \in \DD$, is measured by the structured condition number $\kappa_\DD(A,\lambda)$. Here $\DD$ denotes the set of admissible perturbations. However, if $\DD$ is not a vector space over $\C$ then $\kappa_\DD(A,\lambda)$ provides only incomplete information about the mobility of $\lambda$ under small perturbations from $\DD$. The full information is then given by a certain set $K_\DD(x,y)\subset \C$ which depends on $\DD$ and a pair of normalized right and left eigenvectors $x,y$. In this paper we study the sets $K_\DD(x,y)$ and obtain methods for computing them. In particular we show that $K_\DD(x,y)$ is an ellipse in some important cases.

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Metadaten
Author:Michael Karow
URN:urn:nbn:de:0296-matheon-4192
Referee:Volker Mehrmann
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2007/09/27
Release Date:2007/09/27
Tag:
MSC-Classification:15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY / 15Axx Basic linear algebra / 15A18 Eigenvalues, singular values, and eigenvectors
65-XX NUMERICAL ANALYSIS / 65Fxx Numerical linear algebra / 65F15 Eigenvalues, eigenvectors
65-XX NUMERICAL ANALYSIS / 65Fxx Numerical linear algebra / 65F35 Matrix norms, conditioning, scaling [See also 15A12, 15A60]
Preprint Number:407
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