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Matrices commuting with a given normal tropical matrix. (English) Zbl 1321.15046
Summary: Consider the space $$M_n^{\mathrm{nor}}$$ of square normal matrices $$X = (x_{i j})$$ over $$\mathbb{R} \cup \{- \infty \}$$, i.e., $$- \infty \leq x_{i j} \leq 0$$ and $$x_{i i} = 0$$. Endow $$M_n^{\mathrm{nor}}$$ with the tropical sum $$\oplus$$ and multiplication $$\odot$$. Fix a real matrix $$A \in M_n^{\mathrm{nor}}$$ and consider the set $$\Omega(A)$$ of matrices in $$M_n^{\mathrm{nor}}$$ 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 $$A \odot X = X \odot A = A$$ is also a finite union of alcoved polytopes. The same is true for the set $$\Omega'(A)$$ of $$X$$ such that $$A \odot X = X \odot A = X$$.
A topology is given to $$M_n^{\mathrm{nor}}$$. 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 $$\operatorname{span}A$$, $$\operatorname{span}X$$ and $$\operatorname{span}(A X)$$, when $$A$$ and $$X$$ commute. Two matrices, denoted $$\underline{A}$$ and $$\overline{A}$$, 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.

##### MSC:
 15A80 Max-plus and related algebras 14T05 Tropical geometry (MSC2010) 15B57 Hermitian, skew-Hermitian, and related matrices
TropLi
Full Text:
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