Min-plus methods in eigenvalue perturbation theory and generalised Lidskii-Vishik-Ljusternik theorem

Details Min-plus methods in eigenvalue perturbation theory and generalised Lidskii-Vishik-Ljusternik theorem Min-plus methods in eigenvalue perturbation theory and generalised Lidskii-Vishik-Ljusternik theorem

2004-02-05

arxiv.org/abs/math/0402090v3

Mathematics

Spectral Theory

38 pages, 7 postscript figures, LaTeX2e v1, 5 Feb 04. v2, 20 Jan 05: final example added, more informative title and introduct

Scientific paper

We extend the perturbation theory of Vishik, Ljusternik and Lidskii for eigenvalues of matrices, using methods of min-plus algebra. We show that the asymptotics of the eigenvalues of a perturbed matrix is governed by certain discrete optimisation problems, from which we derive new perturbation formulae, extending the classical ones and solving cases which where singular in previous approaches. Our results include general weak majorisation inequalities, relating leading exponents of eigenvalues of perturbed matrices and min-plus analogues of eigenvalues.

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