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Resumen:
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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 A?Mnnor and consider the set ?(A) of matrices in Mnnor which commute with A. We prove that ?(A) is a finite union of alcoved polytopes; in particular, ?(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 ?A(A) is a neighborhood of the identity matrix I. If A is strictly normal, then ??(A) is a neighborhood of the zero matrix. In one case, ?(A) is a neighborhood of A. We give an upper bound for the dimension of ??(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 ?(A). The geometric meaning of them is given in detail, for one example. We produce examples of matrices which commute, in any dimension.
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