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Probability measures on the space of Hermitian matrices which are ergodic for the conjugation action of an infinite-dimensional unitary group are considered. It is established that the eigenvalues of random matrices distributed with respect to these measures satisfy the law of large numbers. The relationship between such models of random matrices and objects in free probability, freely infinitely divisible measures, is also established. [ABSTRACT FROM AUTHOR]

We obtain a new criterion for the complete indeterminacy of the classical matrix problem and of the Nevanlinna-Pick matrix problem in terms of the convergence of two matrix series. The elements of these series are positive matrices and are expressed by analogs of the classical Schur parameters. [ABSTRACT FROM AUTHOR]

Necessary and sufficient conditions for the Stieltjes moment problem to have a unique solution and for the Hamburger moment problem with the same moments to have infinitely many solutions were obtained in Hamburger's papers on the classical moment problem. In this paper, we obtain an analog of Hamburger's criterion for the matrix moment problem. [ABSTRACT FROM AUTHOR]

In this note, for a system of q matrix equations of the form we consider the problem of the existence of Hermitian nonnegative-definite solutions. We offer an alternative with simplification and regularity to the result on necessary and sufficient conditions for the above matrix equations with q = 2 to have a Hermitian nonnegative-definite solution obtained by Zhang [1], who proposed a revised version of Young et al. [2]. Moreover, we give a necessary condition for the general case and then put forward a conjecture, with which at least some special situations are in agreement. [ABSTRACT FROM AUTHOR]

We show that any n × n conjugate-normal matrix can be brought by a unitary congruence transformation to block-tridiagonal form with the orders of the consecutive diagonal blocks not exceeding 1, 2, 3, ..., respectively. The proof is constructive; namely, a finite process is described that implements the reduction to the desired form. Sufficient conditions are indicated for the orders of the diagonal blocks to stabilize. In this case, the condensed form is a band matrix. [ABSTRACT FROM AUTHOR]

Focuses on the stability of linear stochastic finite-difference systems with almost-periodic coefficients. Reduction of the stability problem of solutions of deterministic linear finite-difference equation; Establishment of a condition for exponential stability; Validity of the criterion of asymptotic mean-square stability.

Solvability conditions are examined for the matrix equation $$X^{ - 1} X* = A$$ , which cannot be found in the well-known reference books on matrix theory. Methods for constructing solutions to this equation are indicated. [ABSTRACT FROM AUTHOR]

We obtain necessary and sufficient conditions for the solvability of the augmentation and modification problems of order $$r$$ for Hermitian matrices. The augmentation problem consists in the construction of a Hermitian $$((n + r) \times (n + r))$$ -matrix with a given $$(n \times n)$$ -block $$A_{11} $$ in block $$(2 \times 2)$$ -representation and with the prescribed eigenvalues. The modification problem consists in the construction of a Hermitian $$(n \times n)$$ -matrix $$B$$ of rank not greater than $$r$$ so that the obtained matrix, being added to a given Hermitian $$(n \times n)$$ -matrix $$A$$ , will have the required spectrum. We give an estimate for the minimal number of different eigenvalues of the solutions to these problems. [ABSTRACT FROM AUTHOR]

We give a simple proof of a determinantal inequality due to Marcus (1976). [ABSTRACT FROM AUTHOR]

The article discusses the measurement of the orthogonal matrix polynomials with periodic coefficients of recurrence relations. It uses the diagonalization of the matrix transform to determine the matrices. It presents mathematical theorems that explain the problem of reconstructing the measure of a periodic sequence of coefficients.