# Matrices commuting with a given normal tropical matrix

### Impacto

Linde, J. and Puente Muñoz, María Jesús de la (2015) Matrices commuting with a given normal tropical matrix. Linear Algebra and its Applications, 482 . pp. 101-121. ISSN 0024-3795

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Official URL: http://dx.doi.org/10.1016/j.laa.2015.04.032

## 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.

Item Type: Article Tropical algebra; Commuting matrices; Normal matrix; Idempotent matrix; Alcoved polytope; Convexity Sciences > Mathematics > Algebra 32489 23 Jul 2015 11:33 20 Jan 2016 15:10