Matrices commuting with a given normal tropical matrix

Consider the space $M_n^{nor}$ of square normal matrices $X=(x_{ij})$ over $\mathbb{R}\cup\{-\infty\}$, i.e., $-\infty\le x_{ij}\le0$ and $x_{ii}=0$. Endow $M_n^{nor}$ with the tropical sum $\oplus$ and multiplication $\odot$. Fix a real matrix $A\in M_n^{nor}$ and consider the set $Ω(A)$ of matrices in $M_n^{nor}$ which commute with $A$. We prove that $Ω(A)$ is a finite union of alcoved polytopes; in particular, $Ω(A)$ is a finite union of convex sets. The set $Ω^A(A)$ of $X$ such that $A\odot X=X\odot A=A$ is also a finite union of alcoved polytopes. The same is true for the set $Ω'(A)$ of $X$ such that $A\odot X=X\odot A=X$. A topology is given to $M_n^{nor}$. Then, the set $Ω^{A}(A)$ is a neighborhood of the identity matrix $I$. If $A$ is strictly normal, then $Ω'(A)$ is a neighborhood of the zero matrix. In one case, $Ω(A)$ is a neighborhood of $A$. We give an upper bound for the dimension of $Ω'(A)$. We explore the relationship between the polyhedral complexes $span A$, $span X$ and $span (AX)$, when $A$ and $X$ commute. Two matrices, denoted $\underline{A}$ and $\bar{A}$, arise from $A$, in connection with $Ω(A)$. The geometric meaning of them is given in detail, for one example. We produce examples of matrices which commute, in any dimension.

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