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The paper proves that an indecomposable nonnegative matrix is chainable if and only if it satisfies Kolmogorov's condition, under which the Markov chain determined by a stochastic matrix obeys the multidimensional local limit theorem. Bibliography: 8 titles., 1. Introduction This paper continues the study of nonnegative chainable matrices initiated in [1]. Familiarity with [1] is desirable, but all the necessary definitions and facts are provided below. Additional [...]

The paper introduces and explores the notions of ergodicity index and ergodicity exponent of a set of stochastic matrices. For the ergodicity exponent a sharp upper bound is obtained. A particular case of this bound is the well-known Paz bound. Also a connection between the ergodicity index and the Protasov–Voynov imprimitivity index is established. [ABSTRACT FROM AUTHOR]

The notion of a temporal component of a semigroup of block-monomial nonnegative matrices is introduced. For such semigroups, a generalization of Minc's theorem on the structure of an irreducible matrix is proved. Bibliography: 8 titles., UDC 512.6 1. INTRODUCTION Recall that a nonnegative matrix A of order n is said to be irreducible if, for arbitrary two indices i, j from the set N = [...]

Generalizations of the Protasov-Voynov theorem on the structure of irreducible semigroups of nonnegative matrices free of zero rows and columns are obtained. The theorem is extended to semigroups that are allowed to be reducible and to matrices that may have zero columns. The main results concern the semigroups called entire. In the definitions and proofs, only combinatorial properties of nonnegative matrices are exploited. Bibliography: 7 titles, UDC 512.6 Introduction In accordance with a well-known Frobenius theorem, an irreducible nonnegative matrix is either primitive (and then a certain power of the matrix has no zero entries) or [...]

The main result of the paper is a theorem, using which a new proof of Roth's theorem is obtained, a new solvability criterion for the matrix equation AX - YB [...]

It is shown that the unitary similarity of two matrix algebras generated by pairs of orthoprojectors {[P.sub.1], [Q.sub.1]} and {[P.sub.2], [Q.sub.2]} can be verified by comparing the traces of [P.sub.1], [...]

The Frobenius form of an irreducible bistochastic matrix is clarified. A generalization of the Frobenius form for irreducible semigroups of bistochastic matrices is described. [ABSTRACT FROM AUTHOR]

A description of nonnegative chainable matrices, based on fully indecomposable matrices, is given. The notion of the chainable rank of a nonnegative matrix is introduced and investigated. Properties of chainable matrices are used in studying stochastic Sarymsakov matrices. [ABSTRACT FROM AUTHOR]

It is proved that the index of imprimitivity of a semigroup of nonnegative block-monomial matrices free of zero rows decomposes into the sum of the indices of imprimitivity of its temporal components, and if the semigroup is block irreducible, then the indices of imprimitivity of all its temporal components coincide. [ABSTRACT FROM AUTHOR]

A new combinatorial proof of the Protasov-Voynov theorem on the structure of irreducible semigroups of nonnegative matrices is proposed. The original proof was obtained by geometric methods. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The criterion for coincidence of spectral radiuses of a complex matrix A and of the matrix | A| obtained by replacing entries of A by their absolute values was given by Wielandt. In this paper we give the new criterion for the above coincidence. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A counterpart of the well-known Harary theorem on signed graphs is proved for digraphs over groups. This result is then used to derive a known theorem on the diagonal similarity of matrices and Kolmogorov's criterion of the reversibility of Markov chains. [ABSTRACT FROM AUTHOR]

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 [ABSTRACT FROM AUTHOR]

Let A and B be square complex matrices of the same order n. Based on an important result Y. P. Hong and R. A. Horn, we propose a criterion for verifying unitary congruence of these matrices. The criterion requires that a finite number of arithmetic operations be performed. No criteria with this finiteness property were previously known. Bibliography: 7 titles. [ABSTRACT FROM AUTHOR]

A formula for the smallest of the ranks of infinite Toeplitz extensions of a finite rectangular Toeplitz matrix is derived. Bibliography:3titles. [ABSTRACT FROM AUTHOR]

The article discusses the use of the classical Specht theorem as an effective criterion for determining the unitary congruence between matrices A and B. The theorem requires the need to verify equalities of infinite set of conditions. It emphasizes that matrices A and B are said to be unitarily similar if there exists a unitary matrix U. A theorem in the theory of unitary congruences indicates that every complex symmetric matrix C can be brought by a unitary congruence transformation to a diagonal form whose diagonal entries are the singular values of matrix A.

The paper suggests a new linear-algebraic proof of a formula for the Perron vector (stationary distribution) of a stochastic matrix, known in the theory of Markov chains. Bibliography: 5 titles. [ABSTRACT FROM AUTHOR]