# On $2\times 2$ Tropical Commuting Matrices

## TL;DR

This paper explores the geometric structure of $2\times 2$ tropical matrices that commute, providing a criterion to determine when two such matrices commute in max linear algebra, with implications for tropical matrix theory.

## Contribution

It characterizes the extremal structure of the tropical polyhedral cone for $2\times 2$ matrices and introduces a new criterion for matrix commutativity in tropical algebra.

## Key findings

- Identified extremals of the tropical polyhedral cone for commuting matrices.
- Developed a criterion to test commutativity of $2\times 2$ tropical matrices.
- Enhanced understanding of the geometric properties of 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.