Matrices commuting with a given normal tropical matrix

Consider the space Mnnor of square normal matrices X=(xij)X=(xij) over R∪{−∞}R∪{−∞}, i.e., −∞≤xij≤0−∞≤xij≤0 and xii=0xii=0. Endow Mnnor with the tropical sum ⊕ and multiplication ⊙. Fix a real matrix A∈Mnnor and consider the set Ω(A)Ω(A) of matrices in Mnnor which commute with A  . We prove that Ω(A)Ω(A) is a finite union of alcoved polytopes; in particular, Ω(A)Ω(A) is a finite union of convex sets. The set ΩA(A)ΩA(A) of X   such that A⊙X=X⊙A=AA⊙X=X⊙A=A is also a finite union of alcoved polytopes. The same is true for the set Ω′(A)Ω′(A) of X   such that A⊙X=X⊙A=XA⊙X=X⊙A=X.A topology is given to Mnnor. Then, the set ΩA(A)ΩA(A) is a neighborhood of the identity matrix I. If A   is strictly normal, then Ω′(A)Ω′(A) is a neighborhood of the zero matrix. In one case, Ω(A)Ω(A) is a neighborhood of A  . We give an upper bound for the dimension of Ω′(A)Ω′(A). We explore the relationship between the polyhedral complexes span A, span X   and span(AX)span(AX), when A and X   commute. Two matrices, denoted A̲ and A¯, arise from A  , in connection with Ω(A)Ω(A). The geometric meaning of them is given in detail, for one example. We produce examples of matrices which commute, in any dimension.