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

2004/02/05 by Marianne Akian, Akian, Marianne, Ravindra Bapat +3
Mathematics · #05C50 #12K10 #47A55 #47A75 #FOS: Mathematics #Spectral Theory (math.SP) #math.SP #msc:05C50 #msc:12K10 #msc:47A55 #msc:47A75

paper · pdf · doi:10.48550/arxiv.math/0402090

38 pages, 7 postscript figures, LaTeX2e v1, 5 Feb 04. v2, 20 Jan 05: final example added, more informative title and introduction. v3, 14 Feb 06: references added

arxiv created 2006/02/16 · arxiv updated 2009/12/01

Abstract

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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