An algorithm for the eigenvalue perturbation problem: reduction of a κ-matrix to a Lidskii matrix

Authors:
Claude-Pierre Jeannerod
Affiliations:
LMC-IMAG, 51 rue des Mathématiques, 38041 Grenoble Cedex 9, France
Venue:
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Year:
2000

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Abstract

In this article, we present an algorithmic approach to the eigenvalue perturbation problem. We show that any matrix perturbation A(&egr;) of an arbitrary nilpotent Jordan canonical form J with all eigenvalues having an order of the form O(&egr;1/(a positive integer)) is similar to a matrix perturbation Atilde;(&egr;) in Arnold normal form that can be seen as generic. Calling A(&egr;) a &kgr;-matrix and Atilde;(&egr;) a Lidskii-Arnold matrix, we also provide a reduction algorithm for the computation of the Lidskii-Arnold form of a &kgr;-matrix. It is based on the minimization of the leading Jordan structure J and on Lidskii's genericity conditions for perturbed eigenvalues.