Journal of Mathematical Sciences

, Volume 224, Issue 6, pp 815–820 | Cite as

Locally Strongly Primitive Semigroups of Nonnegative Matrices

  • Yu. A. Al’pin
  • V. S. Al’pina

The class of locally strongly primitive semigroups of nonnegative matrices is introduced. It is shown that, by a certain permutation similarity, all the matrices of a semigroup of the class considered can be brought to a block monomial form; moreover, any matrix product of sufficient length has positive nonzero blocks only. This shows that the following known property of an imprimitive nonnegative matrix in Frobenius form is inherited: If such a matrix is raised to a sufficiently high power, then all its nonzero blocks are positive. A combinatorial criterion of the locally strong primitivity of a semigroup of nonnegative matrices is found. Bibliography: 6 titles.


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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Kazan (Volga Region) Federal UniversityKazanRussia
  2. 2.Kazan National Research Technological UniversityKazanRussia

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