Classification of $k$-Primitive Sets of Matrices

Abstract

We develop a new approach for characterizing k-primitive matrix families. Such families generalize the notion of a primitive matrix. They have been intensively studied in the recent literature due to applications to Markov chains, linear dynamical systems, and graph theory. We prove, under some mild assumptions, that a set of k nonnegative matrices is either k-primitive or there exists a nontrivial partition of the set of basis vectors, on which these matrices act as commuting permutations. This gives a convenient classification of k-primitive families and a polynomial-time algorithm to recognize them. This also extends some results of Perron-Frobenius theory to nonnegative matrix families.

On primitivity of sets of matrices

“…Other generalizations of the well-studied primitivity of one single matrix to a set of matrices exist in the literature. See for instance [20] or [9], and [23] for a recent paper on so-called k-primitivity.…”