On Primitivity of Sets of Matrices.

A nonnegative matrix A is called primitive if A k 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 = { A 1 , A 2 , ¿ , A m } is primitive if A i 1 A i 2 ¿ A i k is positive for some indices i 1 , i 2 , . . . , i k . 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 = N P 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 ¿erný 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 ( n 3 ) .