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 stochastic matrices and the spectral properties of matrices in the semigroup is presented.
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V. Yu. Protasov and A. S. Voynov, “Sets of nonnegative matrices without positive products,” Linear Algebra Appl., 437, 749–765 (2012).
Yu. A. Al’pin and V. S. Al’pina, “Combinatorial properties of irreducible semigroups of nonnegative matrices,” Zap. Nauchn. Semin. POMI, 405, 13–23 (2012).
Yu. A. Al’pin and V. S. Al’pina, “Combinatorial properties of entire semigroups of nonnegative matrices,” Zap. Nauchn. Semin. POMI, 428, 13–31 (2014).
E. Seneta, Non-Negative Matrices And Markov Chains, Springer, New York (2006).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 439, 2015, pp. 13–25.
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Al’pin, Y.A., Al’pina, V.S. Combinatorial and Spectral Properties of Semigroups of Stochastic Matrices. J Math Sci 216, 730–737 (2016). https://doi.org/10.1007/s10958-016-2936-5
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DOI: https://doi.org/10.1007/s10958-016-2936-5