  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 $\Omega(A)$ 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
$\Omega'(A)$ 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
$\Omega'(A)$ 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
$\Omega'(A)$. 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
$\Omega(A)$. The geometric meaning of them is given in detail, for one example.
We produce examples of matrices which commute, in any dimension.
