Dynamics of Products of Matrices in Max Algebra

TL;DR

This paper investigates the behavior of matrix products in max algebra, extending Perron-Frobenius concepts to understand periodic points and dynamics of finite products from collections of nonnegative matrices.

Contribution

It generalizes Perron-Frobenius theorem results to max algebra and analyzes the dynamics of finite matrix products with bounded maximum circuit geometric mean.

Findings

Extended Perron-Frobenius theorem to max algebra.

Analyzed periodic points of matrix products in max algebra.

Studied dynamics of products from collections of matrices with max circuit mean ≤ 1.

Abstract

The aim of this manuscript is to understand the dynamics of matrix products in a max algebra. A consequence of the Perron-Fr\"{o}benius theorem on periodic points of a nonnegative matrix is generalized to a max algebra setting. The same is then studied for a finite product associated to a $p$-lettered word on $N$ letters arising from a finite collection of nonnegative matrices, with each member having its maximum circuit geometric mean at most 1.