Matrices commuting with a given normal tropical matrix

Consider the space M-n(nor) of square normal matrices X = (x(ij)) over R boolean OR {-infinity}, i.e., -infinity <= x(ij) <= 0 and x(ii) = 0. Endow M-n(nor) with the tropical sum circle plus and multiplication circle dot. Fix a real matrix A is an element of 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 circle dot X = X circle dot A = A is also a finite union of alcoved polytopes. The same is true for the set Omega'(A) of X such that A circle dot X = X circle dot 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 A and (A) over bar, 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. (C) 2015 Elsevier Inc. All rights reserved.