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

TL;DR

This paper extends eigenvalue perturbation theory using min-plus algebra, deriving new asymptotic formulas and inequalities that handle singular cases and relate eigenvalues to discrete optimization problems.

Contribution

It introduces min-plus algebra methods into eigenvalue perturbation theory, providing new formulas and inequalities that generalize classical results and address singular cases.

Findings

Derived new eigenvalue asymptotics governed by discrete optimization.

Extended classical perturbation formulas to singular cases.

Established min-plus analogues of eigenvalues and majorisation inequalities.

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.