Matrices commuting with a given normal tropical matrix

Abstract

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

Journal version

References in corpus (1)

• Computing the vertices of tropical polyhedra using directed hypergraphs