Journal of Mathematical Sciences

, Volume 191, Issue 1, pp 4–9 | Cite as

Combinatorial properties of irreducible semigroups of nonnegative matrices


The paper suggests a combinatorial proof of the Protasov–Voynov theorem on an irreducible semigroup of nanonegative matrices free of positive matrices. This solves the problem posed by the authors of the theorem. Bibliography: 6 titles


Combinatorial Property Nonnegative Matrice Positive Matrice Combinatorial Proof Irreducible Semigroup 
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  1. 1.
    V. Yu. Protasov, “Semigroups of nonnegative matrices,” Usp. Mat. Nauk, 65, 191–192 (2010).MathSciNetCrossRefGoogle Scholar
  2. 2.
    V. Yu. Protasov and A. S. Voynov, "Sets of nonnegative matrices without positive products," Linear Algebra Appl., 437, 749–765 (2012).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    V. Romanovsky, “Un théorème sur les zéros des matrices nonnégatives,” Bull. Soc. Math. Frane, 61, 213–219 (1933).MathSciNetGoogle Scholar
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    V. I. Romanovsky, Diskrete Markov Chains [in Russian], Gostekhizdat, Mosow (1949).Google Scholar
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    H. Min, Nonnegative Matrices, Wiley, New York (1988).Google Scholar
  6. 6.
    Yu. A. Al’pin and V. S. Al’pina, “The Perron-Frobenius theorem — a proof with the use of Markov chains,” Zap. Nauchn. Semin. POMI, 359, 6–16 (2008).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

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

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