On commuting matrices in max algebra and in classical nonnegative algebra

TL;DR

This paper explores the properties of commuting matrices in max algebra and nonnegative linear algebra, extending classical results and analyzing eigenstructure, Frobenius forms, and eigencone intersections.

Contribution

It introduces max algebra analogues of classical matrix results, studies Frobenius normal forms for commuting matrices, and characterizes eigencone intersections and common eigenvectors.

Findings

Existence of common eigenvectors in max algebra and nonnegative algebra.

Max algebra analogues of classical matrix theorems.

Conditions for common eigencones and eigennodes in commuting matrices.

Abstract

This paper studies commuting matrices in max algebra and nonnegative linear algebra. Our starting point is the existence of a common eigenvector, which directly leads to max analogues of some classical results for complex matrices. We also investigate Frobenius normal forms of commuting matrices, particularly when the Perron roots of the components are distinct. For the case of max algebra, we show how the intersection of eigencones of commuting matrices can be described, and we consider connections with Boolean algebra which enables us to prove that two commuting irreducible matrices in max algebra have a common eigennode.