Matrices commuting with a given normal tropical matrix

الأصل · EN

Consider the space Mₙnor of square normal matrices X=(xij) over R∪{-∞}, i.e., -∞≤ xij≤0 and xii=0. Endow Mₙnor with the tropical sum ⊕ and multiplication. Fix a real matrix A∈ Mₙnor 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 Ωᵃ(A) of X such that A X=X A=A is also a finite union of alcoved polytopes. The same is true for the set Ω'(A) of X such that A X=X A=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.