Asymptotic formulae and inequalities for point spectrum in max algebra

TL;DR

This paper develops new asymptotic formulas and inequalities relating max-algebra eigenvalues to classical eigenvalues of nonnegative matrices, facilitating the transfer of spectral results between these frameworks.

Contribution

It introduces explicit asymptotic relations and inequalities for eigenvalues in max-algebra and classical matrix theory, including a spectral mapping theorem for the distinguished spectrum.

Findings

Derived explicit asymptotic formulas linking max-algebra and classical eigenvalues.

Established inequalities for Hadamard products and weighted geometric means of nonnegative matrices.

Presented a spectral mapping theorem for the distinguished spectrum.

Abstract

We prove new explicit asymptotic formulae between (geometric) eigenvalues in max-algebra and classical distinguished eigenvalues of nonnegative matrices, which are useful tools for transferring results between both settings. We establish new inequalities for both types of eigenvalues of Hadamard products and Hadamard weighted geometric means of nonnegative matrices. Moreover, a version of the spectral mapping theorem for the distinguished spectrum is pointed out.