Vol 240, No 6 (2019)

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.

The paper studies the anticommutativity condition for elements of arbitrary real Cayley–Dickson algebras. As a consequence, the anticommutativity graphs on equivalence classes of such algebras are classified. Under some additional assumptions on the algebras considered, an expression for the centralizer of an element in terms of its orthogonalizer is obtained. Conditions sufficient for this interrelation to hold are provided. Also examples of real Cayley–Dickson algebras in which the centralizer and orthogonalizer of an element are not interrelated in this way are considered.

With a square complex matrix A the matrix pair consisting of its symmetric S(A) = (A + A^T)/2 and skew-symmetric K(A) = (A − A^T)/2 parts is associated. It is shown that square matrices A and B are congruent if and only if the associated pairs (S(A), K(A)) and (S(B), K(B)) are (strictly) equivalent. This criterion can be verified by a rational calculation, provided that the entries of A and B are rational or rational Gaussian numbers.

Let K be a nonsingular skew-symmetric matrix of even order n = 2m. For such a matrix, the paper proposes a finite algorithm for computing an m-dimensional neutral subspace, which uses arithmetic operations and quadratic radicals only. The necessity of computing neutral subspaces originates in the problem of solving quadratic matrix equations.

The paper studies two numerical characteristics of matrix incidence algebras over finite fields associated with generating sets of such algebras: the minimal cardinality of a generating set and the length of an algebra. Generating sets are understood in the usual sense, the identity of the algebra being considered a word of length 0 in generators, and also in the strict sense, where this assumption is not used. A criterion for a subset to generate an incidence algebra in the strict sense is obtained. For all matrix incidence algebras, the minimum cardinality of a generating set and a generating set in the strict sense are determined as functions of the field cardinality and the order of the matrices. Some new results on the lengths of such algebras are obtained. In particular, the length of the algebra of “almost” diagonal matrices is determined, and a new upper bound for the length of an arbitrary matrix incidence algebra is obtained.

The paper presents new nonsingularity conditions for n ×n matrices, which involve a subset S of the index set {1, . . ., n} and take into consideration the matrix sparsity pattern. It is shown that the matrices satisfying these conditions form a subclass of the class of nonsingular \( \mathcal{H} \)-matrices, which contains some known matrix classes such as the class of doubly strictly diagonally dominant (DSDD) matrices and the class of Dashnic–Zusmanovich type (DZT) matrices. The nonsingularity conditions established are used to obtain the corresponding eigenvalue inclusion sets, which, in their turn, are used in deriving new inclusion sets for the singular values of a square matrix, improving some recently suggested ones.

The paper computes the Hilbert basis of the cone constructed from matrices describing generic situations, i.e., such vector subspaces in a finite direct sum of finite-dimensional subspaces that are in generic position with respect to the direct summands.