On primitivity of sets of matrices

A nonnegative matrix A is called primitive if ^Ak is positive for some integer k>0. A generalization by Protasov and Voynov (2012) of this concept to finite sets of matrices is as follows: a set of matrices M={^A1,^A2,...,^Am} is primitive if Aⁱ¹A_i2...^A_ik is positive for some indices ⁱ¹,ⁱ²,.,^ik. The concept of primitive sets of matrices comes up in a number of problems within the study of discrete-time switched systems. In this paper, we analyze the computational complexity of deciding if a given set of matrices is primitive and we derive bounds on the length of the shortest positive product. We show that while primitivity is algorithmically decidable, unless P=NP it is not possible to decide primitivity of a matrix set in polynomial time. Moreover, we show that the length of the shortest positive sequence can be superpolynomial in the dimension of the matrices. On the other hand, defining P to be the set of matrices with no zero rows or columns, we give a combinatorial proof of the Protasov-Voynov characterization (2012) of primitivity for matrices in P which can be tested in polynomial time. This latter observation is related to the well-known 1964 conjecture of Černý on synchronizing automata; in fact, any bound on the minimal length of a synchronizing word for synchronizing automata immediately translates into a bound on the length of the shortest positive product of a primitive set of matrices in P. In particular, any primitive set of n×n matrices in P has a positive product of length O(ⁿ³).