On common eigenvectors for semigroups of matrices in tropical and traditional mathematics

TL;DR

This paper proves the existence of a common eigenvector for certain semigroups of matrices across both classical and tropical linear algebra, extending understanding of eigenstructure in these mathematical frameworks.

Contribution

It establishes the existence of common eigenvectors for commutative, nilpotent, and quasinilpotent matrix semigroups in both traditional and tropical settings.

Findings

Existence of common eigenvector for commutative semigroups

Results apply to matrices with real or complex nonnegative entries

Bridges classical and tropical linear algebra theories

Abstract

We prove the existence of a common eigenvector for commutative, nilpotent and quasinilpotent semigroups of matrices with complex or real nonnegative entries both in the conventional and tropical linear algebra.