On $2\times 2$ Tropical Commuting Matrices

Abstract

This paper investigates the geometric properties of a special case of the two-sided system given by $2\times 2$ tropical commuting constraints. Given a finite matrix $A\in {\mathbb{R}}^{2\times 2}$, the paper studies the extremals of the tropical polyhedral cone generated by the entries of matrices $B$ such that $A\otimes B=B\otimes A$ and proposes a criterion to test whether two $2\times 2$ matrices commute in max linear algebra.