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• W1570374530 abstract "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 A̲ 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." @default.
• W1570374530 created "2016-06-24" @default.
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• W1570374530 date "2015-10-01" @default.
• W1570374530 modified "2023-10-01" @default.
• W1570374530 title "Matrices commuting with a given normal tropical matrix" @default.
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• W1570374530 doi "https://doi.org/10.1016/j.laa.2015.04.032" @default.
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