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The paper continues the study of the recently introduced class of [SDD.sub.1] matrices. The class of general [SDD.sub.1] matrices and its three subclasses are considered. In particular, it is shown that [SDD.sub.1] matrices are nonsingular [??]-matrices. Also parameter-free upper bounds for the [l.sub.[infinity]]-norm of the inverses to [SDD.sub.1] matrices are derived. The block triangular form to which any [SDD.sub.1] matrix can be brought by a symmetric permutation of its rows and columns is described. Bibliography: 16 titles., 1. INTRODUCTION AND PRELIMINARIES In 2011, Pena [13] introduced a new matrix class of the so-called [SDD.sub.1] matrices. This class obviously contains the class of Strictly Diagonally Dominant (SDD) matrices [...]

The paper suggests a general approach to deriving upper bounds for [[parallel][A.sup.-1]Q[parallel].sub.[infinity]] from those for [[parallel][A.sup.-1][parallel].sub.[infinity]] for matrices A belonging to different subclasses of the class of nonsingular [??]-matrices. The approach is applied to SDD, S-SDD, OBS, OB, and Nekrasov matrices. Bibliography: 20 titles., 1. INTRODUCTION AND PRELIMINARIES Lately, in a number of papers, some known upper bounds of [[parallel][A.sup.-1][parallel].sub.[infinity]] for n x n matrices A from certain subclasses of nonsingular [??]-matrices were generalized [...]

The paper continues the study of block generalizations of Nekrasov matrices and introduces two new classes of the so-called GN and BJN matrices and compares them with the previously introduced class of GN matrices. Different properties of GN and BJN matrices are established. In particular, it is proved that the classes {GN} and {BJN} are closed with respect to Schur complements and monotone with respect to block partitioning. Also upper bounds for the norms ||[A.sup.-1]||[infinity] of inverses of GN, GN, and BJN matrices A are considered. General results obtained are specialized to the case of block two-by-two matrices with scalar first diagonal block. Bibliography: 19 titles., 1. Introduction and preliminaries The important class of Nekrasov matrices was introduced in [15] and shown to be nonsingular in [7]. Moreover, see [17], every Nekrasov matrix A = ([a.sub.ij]) [...]

The paper considers the classes of GSDD1, GSDD 1 ∗ , and SD-SDD matrices, which contain the class of SDD (strictly diagonally dominant) matrices and are contained in the class of nonsingular H -matrices. New upper bounds on ‖ A - 1 ‖ ∞ for GSDD1, GSDD 1 ∗ , and SD-SDD matrices A, generalizing known upper bounds for S-SDD, SDD 1 ∗ , and GSDD1 matrices, are established and compared. [ABSTRACT FROM AUTHOR]

The paper suggests two general approaches to deriving upper bounds for ‖ A - 1 ‖ ∞ for some eventually H -matrices A. The approaches are illustrated by considering the classes of eventually SDD and eventually DSDD matrices, for which improvements of the known bounds are obtained. Also it is indicated that the approaches suggested are actually applicable to considerably larger matrix classes. [ABSTRACT FROM AUTHOR]

The paper considers some subclasses of the class of nonsingular H-matrices whose definitions involve matrix sparsity pattern. For matrices A in these subclasses, upper bounds for [[parallel][A.sup.-1][parallel].sub.[infinity]] are derived and shown to be sharper than the corresponding bounds ignoring matrix sparsity. Bibliography: 26 titles., UDC 512.643 1. INTRODUCTION In the last years, the problem of bounding the [l.sub.[infinity]] norm of inverse matrices from above has been intensively studied, see, e.g., [4-7, 11-16, 18-22, 24]. [...]

The paper introduces new classes of nonsingular matrices, which contain some known subclasses of the class of nonsingular H-matrices, such as the Nekrasov, Q-Nekrasov, {[P.sub.1], [P.sub.2]}-Nekrasov, and DZ matrices. For matrices in the classes introduced, upper bounds for [[parallel][A.sup.-1][parallel].sub.[infinity]] are derived (in a unified manner) and shown to improve the known bounds for matrices from the corresponding subclasses of the nonsingular H-matrices. Bibliography: 20 titles., UDC 512.643 1. INTRODUCTION Different subclasses of the class of nonsingular H-matrices are intensively studied in the literature, see, in particular, [2-16,20]. Among such classes one should mention the classes [...]

The paper considers the so-called P-Nekrasov and {[P.sub.1], [P.sub.2]}-Nekrasov matrices, defined in terms of permutation matrices P, [P.sub.1], [P.sub.2], which generalize the well-known notion of Nekrasov matrices. For such matrices A, available upper bounds on [[parallel][A.sup.-1][parallel].sub.[infinity]] are recalled, and new upper bounds for the P-Nekrasov and {[P.sub.1], [P.sub.2]}-Nekrasov matrices are suggested. It is shown that the latter bound generally improves the earlier bounds, as well as the bound for the inverse of a P-Nekrasov matrix and the classical bound for the inverse of a strictly diagonally dominant matrix. Bibliography: 12 titles., UDC 512.643 1. INTRODUCTION The paper considers Nekrasov type matrices, to which the Nekrasov, P-Nekrasov, and {[P.sub.1], [P.sub.2]}-Nekrasov matrices are, in particular, referred. Here and below, P and [P.sub.i], i [...]

The paper presents new nonsingularity conditions for n x 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 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. Bibliography: 11 titles., UDC 512.643 1. INTRODUCTION The paper suggests new nonsingularity conditions for square matrices of order n [greater than or equal to] 2, which depend on a nonempty proper subset S [...]

The paper is mainly devoted to studying the so-called Dashnic-Zusmanovich type (DZT) matrices, introduced recently. Interrelations among the DZT matrices and related subclasses of the class of nonsingular H-matrices, namely, the Dashnic-Zusmanovich (DZ) and S-SDD matrices are considered. Upper bounds for the [l.sub.[infinity]]-norms of the inverses to DZT, DZ, and strictly diagonally dominant (SDD) matrices are obtained. A new eigenvalue inclusion set is provided. Bibliography: 17 titles., UDC 512.643 1. INTRODUCTION The class of H-matrices is extremely important in matrix theory. But, in the general case, it is hard to verify whether a given matrix belongs to [...]

The new upper bound [mathematical expression not reproducible] for the largest eigenvalue of a Hermitian positive semidefinite block banded matrix A = ([A.sub.ij]) of block semibandwidth p is suggested. In the special case where the diagonal blocks of A are identity matrices, the latter bound reduces to the bound [[lambda].sub.max](A) [less than or equal to] p + 1, depending on p only, which improves the bounds established for such matrices earlier and extends the bound [[lambda].sub.max](A) [less than or equal to] 2, old known for p = 1, i.e., for block tridiagonal matrices, to the general case p [greater than or equal to] 1. Bibliography: 7 titles., UDC 512.643 The purpose of the present note is to improve known upper bounds for the largest eigenvalue of a Hermitian positive definite (or semidefinite) block banded matrix. Let A [...]

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. Bibliography: 20 titles., UDC 512.643 1. INTRODUCTION AND PRELIMINARIES The paper considers upper bounds for the spectral radii of weighted graph and digraphs. Weighted graphs with positive definite weight matrices were defined in [...]

The paper continues the study of the recently introduced class of SDD1 matrices. The class of general SDD1 matrices and its three subclasses are considered. In particular, it is shown that SDD1 matrices are nonsingular ℌ-matrices. Also parameter-free upper bounds for the l∞-norm of the inverses to SDD1 matrices are derived. The block triangular form to which any SDD1 matrix can be brought by a symmetric permutation of its rows and columns is described. [ABSTRACT FROM AUTHOR]

The paper suggests a general approach to deriving upper bounds for ‖A−1Q‖∞ from those for ‖A−1‖∞ for matrices A belonging to different subclasses of the class of nonsingular ℌ-matrices. The approach is applied to SDD, S-SDD, OBS, OB, and Nekrasov matrices. [ABSTRACT FROM AUTHOR]

The paper suggests a general approach to deriving upper bounds for ‖A−1Q‖∞ from those for ‖A−1‖∞ for matrices A belonging to different subclasses of the class of nonsingular ℌ-matrices. The approach is applied to SDD, S-SDD, OBS, OB, and Nekrasov matrices. [ABSTRACT FROM AUTHOR]

The paper considers upper bounds for the infinity norm of the inverse for matrices in two subclasses of the class of (nonsingular) H-matrices, both of which contain the class of Nekrasov matrices. The first one has been introduced recently and consists of the so-called S-Nekrasov matrices. For S-Nekrasov matrices, the known bounds are improved. The second subclass consists of the socalled QN- (quasi-Nekrasov) matrices, which are defined in the present paper. For QN-matrices, an upper bound on the infinity norm of the inverses is established. It is shown that in application to Nekrasov matrices the new bounds are generally better than the known ones. Bibliography: 15 titles., UDC 512.643 1. INTRODUCTION AND PRELIMINARIES The paper considers two classes of (nonsingular) H-matrices, both containing the Nekrasov matrices, and upper bounds on the infinity norm of the inverse for [...]

Two-sided bounds on [absolute value of det A] for Nekrasov and S-Nekrasov matrices A are obtained. It is shown that for Nekrasov matrices the new bounds improve the known bounds of Bailey and Crabtree. As to the relatively recently introduced S-Nekrasov matrices, no bounds on their determinants have been suggested, as far as the author is aware. Bibliography: 13 titles., UDC 512.643 1. INTRODUCTION In this paper, two-sided bounds on the determinants of the so-called Nekrasov and S-Nekrasov matrices are derived. Matrices of these classes are known to be H-matrices, [...]

It is known that the Nekrasov and S-Nekrasov matrices form subclasses of (nonsingular) Hmatrices. The paper presents some necessary and sufficient conditions for a square matrix with complex entries to be a Nekrasov and an S-Nekrasov matrix. In particular, characterizations of the Nekrasov and S-Nekrasov matrices in terms of the diagonal column scaling matrices transforming them into strictly diagonally dominant matrices are obtained. Bibliography: 15 titles., UDC 512.643 1. INTRODUCTION A matrix A = ([a.sub.ij]) [member of] [C.sup.n x n] is called an H-matrix whenever its comparison matrix M(A) = ([m.sub.ij]), defined by [m.sub.ij] = {[absolute [...]

The paper suggests upper bounds on the second largest eigenvalue and the sum of two largest eigenvalues of symmetric nonnegative matrices and graphs. Conditions necessary and sufficient for some of the bounds to be attained are established. Special attention is paid to the subclass of matrices with zero diagonal entries and with off-diagonal entries not exceeding unity, which obviously contains the adjacency matrices of undirected graphs. Bibliography: 8 titles., 1. Introduction This paper primarily considers the second largest eigenvalue of symmetric nonnegative matrices of order n [greater than or equal to] 2 whose entries do no exceed unity. Following [...]