Matrices commuting with a given normal tropical matrix

Matrices commuting with a given normal tropical matrix

Consider the space M n of square normal matrices X = (xij) over R ∪ {−∞}, i.e., −∞ ≤ xij ≤ 0 and xii = 0. Endow M n with the tropical sum ⊕ and multiplication ⊙. Fix a real matrix A ∈ M n and consider the set Ω(A) of matrices in M n 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 Ω(A) of X such that A...

The upper bound for the index of nilpotency for a matrix commuting with a given nilpotent matrix

We consider the following problem: What are possible sizes of Jordan blocks for a pair of commuting nilpotent matrices? Or equivalently, for which pairs of nilpotent orbits of matrices (under similarity) there exists a pair of matrices, one from each orbit, that commute. The answer to the question could be considered as a generalization of Gerstenhaber– Hesselink theorem on the partial order of...

Commuting triples of matrices

The variety C(3, n) of commuting triples of n × n matrices over C is shown to be irreducible for n = 7. It had been proved that C(3, n) is reducible for n ≥ 30, but irreducible for n ≤ 6. Guralnick and Omladič have conjectured that it is reducible for n > 7.

A Class of Parameter Dependent Commuting Matrices

We present a novel class of real symmetric matrices in arbitrary dimension d, linearly dependent on a parameter x. The matrix elements satisfy a set of nontrivial constraints that arise from asking for commutation of pairs of such matrices for all x, and an intuitive sufficiency condition for the solvability of certain linear equations that arise therefrom. This class of matrices generically vi...

Sampling Real Matrices with Given Margins