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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.

15A80 Max-plus and related algebras
14T05 Tropical geometry (MSC2010)
15B57 Hermitian, skew-Hermitian, and related matrices
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