an:06474476
Zbl 1321.15046
Linde, J.; de la Puente, M. J.
Matrices commuting with a given normal tropical matrix
EN
Linear Algebra Appl. 482, 101-121 (2015).
00347282
2015
j
15A80 14T05 15B57
tropical algebra; commuting matrices; normal matrix; idempotent matrix; alcoved polytope; convexity
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.