卷 232, 编号 6 (2018)

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.

The paper investigates the divisibility of the permanent function of (1,−1)-matrices by different powers of 2. It is shown that the Kräuter–Seifter bound is the best possible one for generic

(1,−1)-matrices.

Two approaches using coarse grid correction in the course of a certain Krylov iterative process are presented. The aim of the correction is to accelerate the iterations. These approaches are based on an approximation of the function sought for by simple basis functions having finite supports. Additional acceleration can be achieved if the iterative process is restarted and the approximate solution is refined. In this case, the resulting process turns out to be a two-level preconditioned method. The influence of different parameters of the iterative process on its convergence is demonstrated by numerical results.

The paper is devoted to studying the orthogonality graph of the matrix ring over a skew field. It is shown that for n ≥ 3 and an arbitrary skew field ????, the orthogonality graph of the ring M_n(????) of n × n matrices over a skew field ???? is connected and has diameter 4. If n = 2, then the graph of the ring M_n(????) is a disjoint union of connected components of diameters 1 and 2. As a corollary, the corresponding results on the orthogonality graphs of simple Artinian rings are obtained.

Adaptive algorithms for constructing spline-wavelet decompositions of matrix flows from a linear space of matrices over a normed field are presented. The algorithms suggested provides for an a priori prescribed estimate of the deviation of the basic flow from the initial one. Comparative bounds of the volumes of data in the basic flow for various irregularity characteristics of the initial flow are obtained in the cases of pseudo-equidistant and adaptive grids. Limit characteristics of the above-mentioned volumes are given in the cases where the initial flow is generated by differentiable functions.

The CMV matrix is the five-diagonal matrix that represents the operator of multiplication by the independent variable in a special basis formed of Laurent polynomials orthogonal on the unit circle C. The article by Cantero, Moral, and Velázquez, published in 2003 and describing this matrix, has attracted much attention because it implies that the conventional orthogonal polynomials on C can be interpreted as the characteristic polynomials of the leading principal submatrices of a certain five-diagonal matrix. The present paper recalls that finite-dimensional sections of the CMV matrix appeared in papers on the unitary eigenvalue problem long before the article by Cantero et al. was published. Moreover, band forms were also found for a number of other situations in the normal eigenvalue problem.

A complete description of the sets of pairs of anti-commuting Toeplitz and Hankel matrices is given.

The paper suggests a general approach to deriving upper bounds for the spectral radii of weighted digraphs. The approach is based on the generalized Wielandt lemma (GWL), which reduces the problem of bounding the spectral radius of a given block matrix to bounding the Perron root of the matrix composed of the norms of the blocks. In the case of the adjacency matrix of weighted graphs and digraphs where all the blocks are square positive (semi)definite matrices of the same order, the GWL takes an especially nice simple form. The second component of the approach consists in applying known upper bounds for the Perron root of a nonnegative matrix. It is shown that the approach suggested covers, in particular, the known upper bounds of the spectral radius and allows one to describe the equality cases.

The following generalization of the theorem on forming a matroid from parts is proved: If a finite set is subdivided into some blocks, each of which is supplied with a matroid structure, and the ranks of every union of certain blocks are prescribed and satisfy the conditions on the rank function of a matroid, then the rank function can be extended to all the subsets of the original set in such a way that the latter becomes a matroid.