Primitivity and Hurwitz Primitivity of Nonnegative Matrix Tuples: A Unified Approach

Abstract

For an -tuple of nonnegative matrices , 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 is the sum of all products with multipliers , . 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 -time algorithm to decide primitivity and an -time algorithm to construct a Hurwitz primitive vector of weight . We also report results on ergodic and Hurwitz ergodic matrix tuples.