Primitive Sets of Nonnegative Matrices and Synchronizing Automata

A set of nonnegative matrices $\mathcal{M}=\{M_1, M_2, \ldots, M_k\}$ is called primitive if there exist possibly equal indices $i_1, i_2, \ldots, i_m$ such that $M_{i_1} M_{i_2} \cdots M_{i_m}$ is entrywise positive. The length of the shortest such product is called the exponent of $\mathcal{M}$. Recently, connections between synchronizing automata and primitive sets of matrices were established. In the present paper, we strengthen these links by providing equivalence results, both in terms of combinatorial characterization and computational complexity. We pay special attention to the set of matrices without zero rows and columns, denoted by $\mathscr{NZ}$, due to its intriguing connections to the Černý conjecture. We rely on synchronizing automata theory to derive a number of results about primitive sets of matrices. Making use of an asymptotic estimate by Rystsov [Cybernetics, 16 (1980), pp. 194--198], we show that the maximal exponent $\exp(n)$ of primitive sets of $n \times n$ matrices satisfy $\lim_{n\rightarrow\infty} \tfrac{\log \exp(n)}{n} = \tfrac{\log 3}{3}$ and that the problem of deciding whether a given set of matrices is primitive is PSPACE-complete, even in the case of two matrices. Furthermore, we characterize the computational complexity of different problems related to the exponent of $\mathscr{NZ}$ matrix sets and present a bound of $2n^2 -5n +5$ on the exponent when considering the subclass of matrices having total support.