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Journal of Mathematical Sciences

, Volume 216, Issue 6, pp 730–737 | Cite as

Combinatorial and Spectral Properties of Semigroups of Stochastic Matrices

  • Yu. A. Al’pinEmail author
  • V. S. Al’pina
Article
  • 32 Downloads

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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References

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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).MathSciNetCrossRefzbMATHGoogle Scholar
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    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).Google Scholar
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    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).Google Scholar
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    E. Seneta, Non-Negative Matrices And Markov Chains, Springer, New York (2006).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Kazan’ Federal UniversityKazan’Russia
  2. 2.Kazan’ National Research Technological UniversityKazan’Russia

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