Search

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 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 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 [...]

The paper refines the classical Ostrowski disk theorem and suggests lower bounds for the smallest-in-modulus eigenvalue and the smallest singular value of a square matrix under certain diagonal dominance conditions. A lower bound for the smallest-in-modulus eigenvalue of a product of m [greater than or equal to] 2 matrices satisfying joint diagonal dominance conditions is obtained. The particular cases of the bounds suggested that correspond to the infinity norm are discussed separately and compared with some known results. Bibliography: 8 titles., UDC 512.643 1. Introduction In 1951, Ostrowski [7] suggested the following generalization of Gerschgorin's disk theorem. Theorem 1.1. Let A = ([a.sub.ij]) [member of] [C.sup.n x n], n[greater than or [...]

The paper suggests sufficient nonsingularity conditions for matrices in terms of certain determinantal relations of diagonal dominance type, which improve and generalize some known results. These conditions are used to describe new eigenvalue inclusion sets and to derive new two-sided bounds on the determinants of matrices satisfying them. Bibliography: 8 titles., UDC 512.643 1. INTRODUCTION This paper considers some sufficient determinantal nonsingularity conditions, which are applicable to matrices that need not be H-matrices. The basic idea is to look for diagonally [...]

Let A = [D.sub.A] + B be a block r x r, r [greater than or equal to] 2, Hermitian matrix of order n, where [D.sub.A] is the block diagonal part of A. The main results of the paper are Theorems 2.1 and 2.2, which state the sharp inequalities [[lambda].sub.1] (A) [greater than or equal to] [[lambda].sub.1] ([D.sub.A] + [xi]B) and [[lambda].sub.n] (A) [less than or equal to] [[lambda].sub.n] ([D.sub.A] + [xi]B), -1/r-1 [less than or equal to] [xi] [less than or equal to] 1, and analyze the equality cases. Some implications of these results are indicated. As applications, matrices occurring in spectral graph theory are considered, and new lower bounds on the chromatic number of a graph are obtained. Bibliography: 7 titles., UDC 512.643 1. Introduction Let A = [([A.sub.ij]).sub.ij=1.sup.r] be a block r x r Hermitian matrix of order n, which is also represented as A = [D.sub.A] + B, where [...]

The paper presents new pseudoblock diagonal-dominance conditions for matrices with fixed block partitioning, which generalize the pointwise kth-order diagonal-dominance conditions and pointwise circuit diagonal-dominance conditions. For matrices satisfying pseudoblock conditions, [...]

The paper presents new upper and lower bounds for the singular values of rectangular matrices explicitly involving the matrix sparsity pattern. These bounds are based on an upper bound for [...]

The paper presents new upper and lower bounds for the Perron root of a nonnegative matrix in terms of the simple circuits of length not exceeding k and the simple [...]

Byline: L. Yu. Kolotilina (1) Abstract: The paper completely solves the problem of optimal diagonal scaling for quasireal Hermitian positive-definite matrices of order 3. In particular, in the most interesting irreducible case, it is demonstrated that for any matrix C from the class considered there is a uniquely determined optimally scaled matrix D .sub.0 .sup.* CD.sub.0 of one of the four canonical types. Formulas for the entries of the diagonal matrix D.sub.0 are presented, as well as formulas for the eigenvalues and eigenvectors of D .sub.0 .sup.* CD.sub.0 and for the optimal condition number of C, which is equal to k(D .sub.0 .sup.* CD.sub.0). The optimality of the Jacobi scaling is analyzed. Bibliography: 10 titles. Author Affiliation: (1) St. Petersburg Department, Steklov Mathematical Institute, St. Petersburg, Russia Article History: Registration Date: 22/11/2005 Received Date: 05/05/2004 Article note: Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 84--126.

Byline: L. Yu. Kolotilina (1) Abstract: The paper considers the singularity/nonsingularity problem for matrices satisfying certain conditions of diagonal dominance. The conditions considered extend the classical diagonal dominance conditions and involve the directed graph of the matrix in question. Furthermore, in the case of the so-called mixed diagonal dominance, the corresponding conditions are allowed to involve both row and column sums for an arbitrary finite set of matrices diagonally conjugated to the original matrix. Conditions sufficient for the nonsingularity of quasi-irreducible matrices strictly diagonally dominant in certain senses are established, as well as necessary and sufficient conditions of singularity/nonsingularity for weakly diagonally dominant matrices in the irreducible case. The results obtained are used to describe inclusion regions for eigenvalues of arbitrary matrices. In particular, a direct extension of the Gerschgorin (r = 1) and Ostrowski-Brauer (r = 2) theorems to r aY= 3 is presented. Bibliography: 18 titles. Author Affiliation: (1) St. Petersburg Department, Steklov Mathematical Institute, St. Petersburg, Russia Article History: Registration Date: 22/11/2005 Received Date: 06/01/2004 Article note: Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 40--83.

A new upper bound on the infinity norm of the inverse to a Nekrasov matrix, improving two recent bounds by L. Cvetkovi'c et al. (Appl. Math. Comput., 219, 5020-5024 (2013)), is obtained. Also it is shown that for a strictly diagonally dominant matrix, the bound suggested improves the classical Varah upper bound. [ABSTRACT FROM AUTHOR]

New characterizations of the so-called PM-and PH-matrices in terms of generalized diagonal dominance are obtained and used to derive some of their properties. Also it is proved that a PH-matrix necessarily has a certain number of strictly diagonally dominant rows. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

Lower and upper conditional bounds for the eigenvalues of a Hermitian positive-definite block 2 × 2 matrix, describing their closest possible location, are derived. The notions of partially and totally optimal spectra are introduced, and several equivalent characterizations of matrices with partially and totally optimal spectra are presented. It is shown that the block 2 × 2 block Jacobi scaled matrices and the so-called equilibrated matrices have totally optimal spectra. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The paper suggests two-sided, upper, and lower circuit bounds for the Perron root of a nonnegative matrix, most of which are derived based on an extension of the monotonicity property of the Perron root established by Fiedler and Pták. Bibliography:9titles. [ABSTRACT FROM AUTHOR]

The lower boundfor the difference of the extreme eigenvalues of an n × n Hermitian block 2 × 2 matrix A =is established, and conditions necessary and sufficient for this bound to be attained at A are provided. Some corollaries of this result are derived. In particular, for a positive-definite matrix A, it is demonstrated that ?1 (A) - ?n (A) = 2?A12? if and only if A is optimally conditioned, and explicit expressions for the extreme eigenvalues of such matrices are obtained. Bibliography:5titles. [ABSTRACT FROM AUTHOR]

The strengthened versions of the classical additive and multiplicative Weyl inequalities for the singular values of A + B and AB*, where A and B are rectangular matrices, and for the eigenvalues of A + B and AB, where A and B are Hermitian matrices, are established under certain assumptions on the subspaces spanned by some singular vectors or eigenvectors, respectively, of A and B. Bibliography:6titles. [ABSTRACT FROM AUTHOR]

A formula for the smallest of the ranks of infinite Toeplitz extensions of a finite rectangular Toeplitz matrix is derived. Bibliography:3titles. [ABSTRACT FROM AUTHOR]

A block 2 × 2 Hermitian positive-definite (h.p.d.) matrix is called equilibrated if its diagonal blocks coincide with the corresponding blocks of its inverse. It is demonstrated that any block 2 × 2 h.p.d. matrix is block diagonally similar to an equilibrated matrix, and any equilibrated matrix is optimally conditioned. Other properties of equilibrated matrices are also established. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

In this paper, bounds and inequalities for the Perron root of a nonnegative matrix, extending and complementing the classical theorems of Frobenius and Ostrowski in terms of (deleted) row and column sums, are presented. All the results are derived by using the same approach, based on the monotonicity property of the Perron root. Bibliography: 12 titles. [ABSTRACT FROM AUTHOR]

This paper studies irreducible matrices $A=(a\_\{ij\})\backslash in\backslash Bbb\; C\backslash ,^\{n\backslash times\; n\}$, $n\backslash ge2$, satisfying Brualdi's conditions $\backslash prod\backslash limits\_\{i\backslash in\backslash overline\backslash gamma\}|a\_\{ii\}|\backslash ge\backslash prod\backslash limits\_\{i\backslash in\backslash overline\; \backslash gamma\}\; R\_i(A),\backslash quad\; \backslash gamma\backslash in\backslash goth\; C(A),$ or, shortly, Brualdi matrices. Here, $R\_i(A)=\backslash sum\backslash limits\_\{j\backslash ne\; i\}|a\_\{ij\}|$, $i=1,\backslash ldots,n$; $\backslash goth\; C(A)$ is the set of circuits of length $k\backslash ge2$ in the directed graph of A, and $\backslash overline\backslash gamma$ is the support of $\backslash gamma$. Among the results obtained are a characterization of Brualdi matrices, implying, in particular, that they are generalized diagonally dominant; the necessary and sufficient conditions of singularity of Brualdi matrices; explicit expressions for the absolute values of the components of right null-vectors of a singular Brualdi matrix, and conditions necessary and sufficient for a boundary point of Brualdi's inclusion region to be an eigenvalue of an irreducible matrix. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

This paper provides a description of block 2 × 2 Hermitian positive-definite matrices that are optimally conditioned with respect to block diagonal similarity transformations. Bibliography: 7 titles. [ABSTRACT FROM AUTHOR]

Let $A^\{(l)\}$$(l=1,\backslash ldots,k)$ be $n\backslash times\; n$ nonnegative matrices with right and left Perron vectors $u^\{(l)\}$ and $v^\{(l)\}$, respectively, and let $D^\{(l)\}$ and $E^\{(l)\}$$(l=1,\backslash ldots,k)$ be positive-definite diagonal matrices of the same order. Extending known results, under the assumption that $u^\{(1)\}\backslash circ\; v^\{(1)\}=\backslash ldots=u^\{(k)\}\backslash circ\; v^\{(k)\}\backslash ne0$ (where ``$\backslash circ$'' denotes the componentwise, i.e., the Hadamard product of vectors) but without requiring that the matrices $A^\{(l)\}$ be irreducible, for the Perron root of the sum $\backslash sum^k\_\{l=1\}D^\{(l)\}A^\{(l)\}E^\{(l)\}$ we derive a lower bound of the form $\backslash rho\backslash left(\backslash sum^k\_\{l=1\}D^\{(l)\}A^\{(l)\}E^\{(l)\}\backslash right)\backslash ge\; \backslash sum^\{k\}\_\{l=1\}\backslash be\_l\backslash rho(A^\{(l)\}),\backslash quad\; \backslash beta\_l>0.$ Also we prove that, for arbitrary irreducible nonnegative matrices $A^\{(l)\}$$(l=1,\backslash ldots,k)$, $\backslash rho\backslash left(\backslash sum^\{k\}\_\{l=1\}A^\{(l)\}\backslash right)\backslash ge\; \backslash sum^k\_\{l=1\}\backslash alpha\_l\backslash rho(A^\{(l)\}),$ where the coefficients $\backslash alpha\_l>0$ are specified using an arbitrarily chosen normalized positive vector. The cases of equality in both estimates are analyzed, and some other related results are established. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

This note provides a description of all those pairs of nonzero vectors $x,y\backslash in\{\backslash Bbb\; C\}^n$, $n\backslash ge2$, for which the generalized Wielandt inequality $|x^*Ay|^2\backslash le\backslash bigg[\backslash frac\{\backslash lambda\_1-\backslash lambda\_n+(\backslash lambda\_1+\backslash lambda\_n)|\backslash varphi|\}\; \{\backslash lambda\_1+\backslash lambda\_n+(\backslash lambda\_1-\backslash lambda\_n)|\backslash varphi|\}\backslash bigg]^2x^*Ax\backslash ,\backslash ,y^*Ay,\; \backslash \; \backslash \; \backslash varphi=\backslash frac\{x^*y\}\{\backslash |x\backslash |\backslash ,\backslash |y\backslash |\},$ where $A\backslash in\{\backslash Bbb\; C\}^\{n\backslash times\; n\}$ is a Hermitian positive-definite matrix with eigenvalues $\backslash lambda\_1\backslash ge\backslash lambda\_2\backslash ge\backslash cdots\backslash ge\backslash lambda\_n$ such that $\backslash lambda\_1>\backslash lambda\_n$, holds with equality. Bibliography: 3 titles. [ABSTRACT FROM AUTHOR]

This paper continues the theoretical and numerical study of the so-called factorized sparse approximate inverse (FSAI) preconditionings of symmetric positive-definite matrices and considers two new approaches to improving them. The first one is based on the a posteriori sparsification of an already constructed FSAI preconditioner, whereas the second one amounts to applying another FSAI preconditioning to an already preconditioned matrix. Numerical results for sample FE problems from structural mechanics are presented. Copyright © 1999 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]