An iterative criterion for H-Matrices

Applied Mathematics and Computation, 2008

It is well-known [D. Carlson, T. Markham, Schur complements of diagonally dominant matrices, Czech. Math. J. 29 (104) (1979) 246-251, [1]] that the Schur complement of a strictly diagonally dominant matrix is strictly diagonally dominant. Also, if a matrix is an H-matrix, then its Schur complement is an H-matrix, too [J. Liu, Y. Huang, Some properties on Schur complements of H-matrices and diagonally dominant matrices, Linear Algebra Appl. 389 (2004) 365-380, [8]]. Recent research showed that the same type of statement holds for some special subclasses of H-matrices, see, for example, Liu et al. [J. Liu, Y. Huang, F. Zhang, The Schur complements of generalized doubly diagonally dominant matrices, Linear Algebra Appl. 378 (2004) 231-244]. The aim of this paper is to show that the proof of these results can be significantly simplified by using ''scaling'' approach as in Zhang et al. [F. Zhang et al., The Schur Complement and its Applications, Springer, New York, 2005] and to give another invariance result of this type.

Special Matrices

A weakly diagonally dominant matrix may or may not be invertible. We characterize, in terms of combinatorial structure and sign pattern when such a matrix is invertible, which is the common case. Examples are given.

Numerical Linear Algebra With Applications, 2017

The problem of determining matrix inertia is very important in many applications, for example, in stability analysis of dynamical systems. In the (point-wise) H-matrix case, it was proven that the diagonal entries solely reveal this information. This paper generalises these results to the block H-matrix cases for 1, 2, and ∞ matrix norms. The usefulness of the block approach is illustrated on 3 relevant numerical examples, arising in engineering. KEYWORDS inertia, eigenvalues, stability, block matrices, H-matrices 1 INTRODUCTION Matrix property, called inertia, is widely studied due to its connection with stability of dynamical systems. For a given complex matrix A, it is defined as the triple in(A) = (n + , n 0 , n −), where n + is the number of eigenvalues of A, which have positive real part, n − is the number of eigenvalues of A, which have negative real part, and n 0 is the dimension of the kernel of A. A special case of matrix inertia in(A) = (n, 0, 0) is known as the stability property of continuous linear dynamical systems, which is very important in many applications in engineering, ecology, medicine, etc. On the other hand, in a broad range of applications, such as economics, population dynamics, and communication and network theory, a concept of diagonal dominance and, more broadly, concept of an H-matrix lie in the basis of a model. 1 For real H-matrices, it is known that their inertia is completely described by the position of its diagonal entries, see Kostić. 2 In this paper, we generalize this result to block H-matrices, following the partitioning approach given in one study 3 and considering cases of 1, 2, and infinity norm. Some results regarding such block H-matrices and infinity norm estimates of their inverses can be found in previous studies 4-7 and Ostrowski. 8 The paper is organized as follows. This section is a collection of preliminary material on (point-wise) strictly diagonally dominant (SDD) and H-matrices and related inertia results. Section 2 starts with the concept of block SDD and block H-matrices and contains the main results of this paper. Finally, the last section discusses relationship between block-and point-wise cases, using numerical examples, arising in applications, to point out the benefits. Throughout the paper, we denote by N ∶= {1, 2, … , n} the set of indices. For a given matrix A = [a kj ] ∈ C n,n , we define r k (A) ∶= ∑ j∈N∖{k} |a kj |, k ∈ N. The point-wise classes of nonsingular matrices that we generalize to the block case are well-known SDD and H-matrix classes: Definition 1. A matrix A = [a kj ] ∈ C n,n is SDD if |a kk | > r k (A) for all k ∈ N.

Linear Algebra and its Applications, 1990

Ahac and Olesky have shown that by using the most diagonaldominat column as a pivot element for M-matrices was numerically stable. The purpose of this paper is to show that this method of selecting a pivoting is stable when finding the incomplete factorization of an H-matrix.

Applied Mathematics and Computation, 2009

a b s t r a c t It is well known, see [D. Carlson, T. Markham, Schur complements of diagonally dominant matrices, Czech. Math. J. 29 (104) (1979) 246-251 [2]; J. Liu, J. Li, Z. Huang, X. Kong, Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices, Linear Alg. Appl. 428 (2008) 1009-1030] [14], that the Schur complement of a strictly diagonally dominant matrix is strictly diagonally dominant, as well as its diagonal-Schur complement. Also, if a matrix is an H-matrix, then its Schur complement and diagonal-Schur complement are H-matrices, too, see [J. Liu, Y. Huang, Some properties on Schur complements of H-matrices and diagonally dominant matrices, Linear Alg. Appl. 389 (2004) 365-380] [13]. Recent research, see [J. Liu, Y. Huang, F. Zhang, The Schur complements of generalized doubly diagonally dominant matrices, Linear Alg. Appl. 378 (2004) 231-244 [12]; J. Liu, J. Li, Z. Huang, X. Kong, Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices, Linear Alg. Appl. 428 (2008) 1009-1030] [14]

Proceedings of the American Mathematical Society, 1970

This article concerns dominance conditions for an nY.m matrix. In the simplest kind of dominance, the (absolute) value of the diagonal element exceeds the sum of the absolute values of the nondiagonal elements on the same row. This condition has been generalized in the literature in several ways, of which we consider ways in which the rows of the matrix cooperate. Our work amounts to a sorting out of certain dominance conditions that belong to a class 6 of dominance conditions. We prove a theorem characterizing all true statements of the form

2020

H−matrices play an important role in the theory and applications of Numerical Linear Algebra. So, it is very useful to know whether a given matrix A ∈ C n,n , usually the coefficient of a complex linear system of algebraic equations or of a Linear Complementarity Problem (A ∈ R n,n , with a ii > 0 for i = 1, 2, . . . , n in this case), is an H−matrix; then, most of the classical iterative methods for the solution of the problem at hand converge. In recent years the set of H−matrices has been extended to what is now known as the set of General H−matrices, and a partition of this set in three different classes has been made. The main objective of this work is to develop an algorithm that will determine the H−matrix character and will identify the class to which a given matrix A ∈ C n,n belongs; in addition, some results on the classes of general H−matrices and a partition of the non-H−matrices set are presented.

2003

In this paper, several new simple criteria for nonsingular H-matrices are obtained by making use of elements of matrices only. Advantages are illustrated by numerical examples. And a necessary condition for nonsingular H-matrices is also presented.

2009

In this paper, we consider properties of the Perron complements of diagonally dominant matrices and H-matrices.

Electronic Journal of Linear Algebra, 2009

An M ∨ -matrix has the form A = sI -B, where s ≥ ρ(B) ≥ 0 and B is eventually nonnegative; i.e., B k is entrywise nonnegative for all sufficiently large integers k. A theory of M ∨ -matrices is developed here that parallels the theory of M-matrices, in particular as it regards exponential nonnegativity, spectral properties, semipositivity, monotonicity, inverse nonnegativity and diagonal dominance.