Combinatorial and spectral properties of semigroups of stochastic matrices

The paper studies the notion of imprimitivity index of a semigroup of nonnegative matrices, introduced by Protasov and Voynov. A new characterization of the imprimitivity index in terms of the scrambling rank of a nonnegative matrix is suggested. Based on this characterization, an independent combinatorial proof of the Protasov–Voynov theorem on the interrelation between the imprimitivity index of a semigroup of stohastic matrices and the spectral properties of matrices in the semigroup is presented.