2012/09/04 by J. Linde, Linde, J., M. J. de la Puente +1 · 1 citation
Computer Science · Mathematics · Engineering · #Polynomial and algebraic computation #Advanced Combinatorial Mathematics #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1209.0660
Consider the space Mnnor of square normal matrices X=(xij) over ℝ∪\-∞\, i.e., -∞≤ xij≤0 and xii=0. Endow Mnnor with the tropical sum ⊕ and multiplication \odot. Fix a real matrix A∈ Mnnor and consider the set Ω(A) of matrices in Mnnor which commute with A. We prove that Ω(A) is a finite union of alcoved polytopes; in particular, Ω(A) is a finite union of convex sets. The set ΩA(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 Ω'(A) of X such that A\odot X=X\odot A=X. A topology is given to Mnnor. Then, the set ΩA(A) is a neighborhood of the identity matrix I. If A is strictly normal, then Ω'(A) is a neighborhood of the zero matrix. In one case, Ω(A) is a neighborhood of A. We give an upper bound for the dimension of Ω'(A). We explore the relationship between the polyhedral complexes span A, span X and span (AX), when A and X commute. Two matrices, denoted \underlineA and A, arise from A, in connection with Ω(A). The geometric meaning of them is given in detail, for one example. We produce examples of matrices which commute, in any dimension.