Matrices commuting with a given normal tropical matrix

Abstract

Consider the space Mnnor of square normal matrices X=(xij) over R∪{-∞}, i.e., -∞≤xij≤0 and ;bsupesup&=0. Endow Mnnor with the tropical sum ⊕ and multiplication. Fix a real matrix AεMnnor and consider the set Ω(A) of matrices in Mnnor 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 ;bsupA;esup&(A) of X such that AX=XA=A is also a finite union of alcoved polytopes. The same is true for the set ′(A) of X such that AX=XA=X. A topology is given to Mnnor. Then, the set ΩA(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 A and 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.