The tropical commuting variety

We study tropical commuting matrices from two viewpoints: linear algebra and algebraic geometry. In classical linear algebra, there exist various criteria to test whether two square matrices commute. Similarly, in the realm of tropical linear algebra, we determine conditions for two tropical matrices that are Kleene stars to commute. Shifting to an algebro-geometric perspective, we explicitly compute the tropicalization of the classical variety of commuting matrices in dimensions 2 and 3.