PRIMITIVITY AND HURWITZ PRIMITIVITY OF NONNEGATIVE MATRIX TUPLES: A UNIFIED APPROACH

For an m-tuple of nonnegative n\times n matrices (A1,...,Am), primitivity/Hurwitz primitivity means the existence of a positive product/Hurwitz product, respectively (all products are with repetitions permitted). The Hurwitz product with a Parikh vector a = (a1, . . . , am) \in Z\geqm0 is the sum of all products with ai multipliers Ai, i = 1, ... , m. Ergodicity/Hurwitz ergodicity means the existence of the corresponding product with a positive row. We give a unified proof for the Protasov-Vonyov characterization (2012) of primitive tuples of matrices without zero rows and columns and for the Protasov characterization (2013) of Hurwitz primitive tuples of matrices without zero rows. By establishing a connection with synchronizing automata, we, under the aforementioned conditions, find an O(n2m)-time algorithm to decide primitivity and an O(n3m2)-time algorithm to construct a Hurwitz primitive vector a of weight \summi=1 ai = O(n3). We also report results on ergodic and Hurwitz ergodic matrix tuples.