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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
Software:
TropLi
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