Volume 224, Nº 6 (2017)

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.

The classical Hurwitz theorem claims that there are exactly four normed algebras with division: the real numbers (ℝ), complex numbers (ℂ), quaternions (ℍ), and octonions (????). The length of ℝ as an algebra over itself is zero; the length of ℂ as an ℝ-algebra equals one. The purpose of the present paper is to prove that the lengths of the ℝ-algebras of quaternions and octonions equal two and three, respectively.

ConsiderCn as the pseudo-unitary space with the inner product defined by the matrix\( {p}_n=\left(\begin{array}{l}\kern4em 1\hfill \\ {}\kern3em 1\hfill \\ {}\kern1em \dots \hfill \\ {}1\hfill \end{array}\right) \).

In this space, centro-unitary matrices play the role of unitary operators.

The main result of this paper is a factorization of an arbitrary centro-unitary matrix of even order into a product of simpler centro-unitary matrices. This result is an implication of a similar fact concerning factorizations of pseudo-unitary matrices of the type (n, n).

Let a complex matrix A be the direct sum of its square submatrices B and C that have no common eigenvalues. Then every matrix X belonging to the centralizer of A has the same block diagonal form as the matrix A. This paper discusses how the conditions on the submatrices B and C should be modified for an analogous assertion about the congruent centralizer of A, which is the set of matrices X such that X^*AX = A, to be valid. Also the question whether the matrices in the congruent centralizer are block diagonal if A is a block anti-diagonal matrix is considered. Bibliography: 2 titles.

The paper presents a generalized block version of the Induced Dimension Reduction (IDR) methods in comparison with the Multi–Preconditioned Semi-Conjugate Direction (MPSCD) algorithms in Krylov subspaces with deflation and low-rank matrix approximation. General and individual orthogonality and variational properties of these two methodologies are analyzed. It is demonstrated, in particular, that for any sequence of Krylov subspaces with increasing dimensions there exists a sequence of the corresponding shrinking subspaces with decreasing dimensions. The main conclusion is that the IDR procedures, proposed by P. Sonneveld and other authors, are not an alternative to but a further development of the general principles of iterative processes in Krylov subspaces.

The paper introduces new subclasses, called P\( \mathrm{\mathscr{H}} \)N(π) and P\( \mathrm{\mathscr{H}} \)QN(π), of (nonsingular) \( \mathrm{\mathscr{H}} \)-matrices of order n dependent on a partition π of the index set {1, . . ., n}, which generalize the classes P\( \mathrm{\mathscr{H}} \)(π), introduced previously, and contain, in particular, such subclasses as those of strictly diagonally dominant (SDD), Nekrasov, S-SDD, S-Nekrasov, QN, and P\( \mathrm{\mathscr{H}} \)(π) matrices. Properties of the matrices introduced are studied, and upper bounds on their inverses in l_∞ norm are obtained. Block generalizations of the classes P\( \mathrm{\mathscr{H}} \)N(π) and P\( \mathrm{\mathscr{H}} \)QN(π) in the sense of Robert are considered.

Also a general approach to defining subclasses \( {\mathcal{K}}^{\pi } \) of the class \( \mathrm{\mathscr{H}} \) containing a given subclass \( \mathcal{K} \) ⊂ \( \mathrm{\mathscr{H}} \) and dependent on a partition π is presented.

The paper considers the problem on the dimensions of intersections of a subspace in the direct sum of a finite series of finite-dimensional vector spaces with sums of pairs of direct summands, provided that the subspace intersection with each of these direct summands is trivial. The problem naturally splits into finding conditions for the existence and representability of the corresponding matroid. The following theorem is proved: If the ranks of all the unions of a series of blocks satisfying the condition on the ranks of subsets in the matroid are given and the blocks have full rank, then this partial rank function may be extended to a full rank function for all the subsets of the base set (the union of all the blocks). Necessary and sufficient conditions on the dimensions of the direct summands and intersections mentioned above for the corresponding matroid to exist are obtained in the case of five direct summands. Bibliography: 5 titles.

The paper establishes the existence of an element with nilpotency index n − 1 in an arbitrary nilpotent commutative subalgebra with nilpotency index n−1 in the algebra of upper niltriangular matrices N_n(????) over a field ???? with at least n elements for all n ≥ 5, and also, as a corollary, in the full matrix algebra M_n(????). The result implies an improvement with respect to the base field of known classification theorems due to D. A. Suprunenko, R. I. Tyshkevich, and I. A. Pavlov for algebras of the class considered.