For an \(m\)-tuple of nonnegative \(n \times n\) matrices \((A_1, \ldots,A_m)\), primitivity and (Hurwitz primitivity) means the existence of a positive product (Hurwitz product). Note that all products are with repetitions allowed. The Hurwitz product with a Parikh vector \(\alpha=(\alpha_1,\ldots,\alpha_m) \in \mathbb{Z}_{\geq 0}^m\) is the sum of all products with \(\alpha_i\) multipliers \(A_i\), \(i=1,2,\ldots,m\). On the other hand, ergodicity (Hurwitz ergodicity) means the existence of the corresponding product (Hurwitz product) with a positive row.
In this manuscript, the authors use the following notation: \(\mathrm{Mat}_n(\mathbb{R}_{\geq 0})\) is the set of \(n \times n\) nonnegative real matrices, while the set of nonnegative matrices that has no zero rows (columns) is denoted by \(NZ_1\) (\(NZ_2\)). For any real number \(x\), \([x]\) denotes the set \(\{i \in \mathbb{N}: i \leq x \}\).
In 2012V. Yu. Protasov andA. S. Voynov [Linear Algebra Appl. 437, No. 3, 749–765 (2012;Zbl 1245.15033)]characterized the primitive tuples of matrices without zero rows and columns and one year laterV. Yu. Protasov [SIAM J. Matrix Anal. Appl. 34, No. 3, 1174–1188 (2013;Zbl 1281.15039)] classified the Hurwitz primitive tuples of matrices without zero rows. These characterizations can be stated as follows:
Theorem. Let \(\mathcal{A} \subseteq\mathrm{Mat}_n(\mathbb{R}_{\geq 0})\) be an irreducible set of matrices belonging to \(NZ_1\). Then \(\mathcal{A}\) is not Hurwitz primitive if and only if we can find a nontrivial partition \(\pi\) of \([n]\) and \(\sigma_A \in \mathrm{Sym}_{|\pi|}\) for all \(A \in \mathcal{A}\) such that \(\sigma_A \sigma_B=\sigma_B \sigma_A\) and \(A\) acts on \(\pi\) subordinate to \(\sigma_A\), for all \(A,B \in\mathcal{A}\).
Theorem. Let \(\mathcal{A} \subseteq\mathrm{Mat}_n(\mathbb{R}_{\geq 0})\) be an irreducible set of matrices belonging to \(NZ_2\). Then \(\mathcal{A}\) is primitive if and only if there is no nontrivial partition \(\pi\) of \([n]\) which is preserved by all elements of \(\mathcal{A}\).
In this paper, the authors give a unified proof of these results. First, adopting the usual approach in combinatorial matrix theory, they explain how to deal with various reachability properties of nonnegative matrix tuples as combinatorial problems about digraphs. Next, the authors present a sketch of a proof of the previous results from the viewpoint of a stable relation. Finally, the two last sections of the paper are devoted to recognition algorithms, finding certifying products, and estimating exponents for various reachability properties of matrix sets from \(NZ_1\) and \(NZ_2\).

Reviewer: Juan Ramón Torregrosa Sánchez (Valencia)

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.