Combinatorial and spectral properties of semigroups of stochastic matrices (Q308663)

Let \(P\) be a multiplicative semigroup of nonnegative matrices of degree \(n\). The notion of the index of imprimitivity of the semigroup was introduced in [\textit{V. Yu. Protasov} and \textit{A. S. Voynov}, Linear Algebra Appl. 437, No. 3, 749--765 (2012; Zbl 1245.15033)]. The article 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 stochastic matrices and the spectral properties of matrices in the semigroup is presented.