On k-primitivity of two classes of digraphs

Gao, Yubin; Shao, Yanling

doi:10.1016/j.laa.2009.01.008

Cited by 1 publication

(2 citation statements)

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“…There are several ways to extend this concept to families of nonnegative matrices, possibly, not commuting (see [4,15,16,19] and references therein). In this paper we deal with k-primitive families introduced by Fornasini and Valcher in [9] and studied in [2,10,11,14,17,21]. We present a new method to characterize and classify such families and show that the k-primitivity can be determined in polynomial time.…”

Section: Introductionmentioning

confidence: 99%

“…The concept of k-primitivity is applied in the study of inhomogeneous Markov chains, multivariate generalizations of Markov chains (see [8] and references therein), linear switching systems, in particular, two-dimensional (2D) systems [9,11,17], graphs and multigraphs [3,2,13,14], etc. In case of one matrix (k = 1) this notion reduces to usual primitivity.…”

Section: Introductionmentioning

confidence: 99%

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Classification of $k$-Primitive Sets of Matrices

Protasov¹

2013

SIAM J. Matrix Anal. & Appl.

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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.

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