On matrix perturbations with minimal leading Jordan structure

Authors:
Claude-Pierre Jeannerod
Affiliations:
INRIA-Laboratoire LIP-Ecole Normale Supérieure de Lyon, 46 Allée d'Italie, 69364 Lyon Cedex 07, France
Venue:
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on linear algebra and arithmetic, Rabat, Morocco, 28-31 May 2001
Year:
2004

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Abstract

We show that any matrix perturbation of an n × n nilpotent complex matrix is similar to a matrix perturbation whose leading coefficient has minimal Jordan structure. Additionally, we derive the property that, for matrix perturbations with minimal leading Jordan structure, the sufficient conditions of Lidskii's perturbation theorem for eigenvalues are necessary too. It is further shown how minimality can be obtained by computing a similarity transform whose entries are polynomials of degree at most n. This relies on an extension of both Lidskii's theorem and its Newton diagram-based interpretation.