Further results on H-matrices and their Schur complements

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]

2009

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

Numerical Linear Algebra with Applications, 2009

It is well known that the Schur complement of some H‐matrices is an H‐matrix. In this paper, the Schur complement of any general H‐matrix is studied. In particular, it is proved that the Schur complement, if it exists, is an H‐matrix and the class to which the Schur complement belongs is studied. In addition, results are given for singular irreducible H‐matrices and for the Schur complement of nonsingular irreducible H‐matrices. Copyright © 2009 John Wiley & Sons, Ltd.

Linear Algebra and its Applications, 2008

Let M(A) denote the comparison matrix of a square H-matrix A, that is, M(A) is an M-matrix. H-matrices such that their comparison matrices are non-singular are well studied in the literature. In this paper, we study characterizations of H-matrices with singular or nonsingular comparison matrix. In particular, we analyze the case when A is irreducible and then give insights into the reducible case. The spectral radius of the Jacobi matrix of M(A) and the generalized diagonal dominance property are used in the characterizations. Finally, from these characterizations, a partition of the general H-matrix set in three classes is obtained.

Linear Algebra and its Applications, 2011

In this paper, we analyze the relation between some classes of matrices with variants of the diagonal dominance property. We establish a sufficient condition for a generalized doubly diagonally dominant matrix to be invertible. Sufficient conditions for a matrix to be strictly generalized diagonally dominant are also presented. We provide a sufficient condition for the invertibility of a cyclically diagonally dominant matrix. These sufficient conditions do not assume the irreducibility of the matrix.

Applied Mathematics and Computation, 2009

Class of H-matrices plays an important role in various scientific disciplines, in economics, for example. However, this class could be used in order to get various benefits in other linear algebra fields, like determinant estimation, Perron root estimation, eigenvalue localization, improvement of convergence area of relaxation methods, etc. For that reason, it seems important to find a subclass of H-matrices, as wide as possible, and expressed by explicit conditions, involving matrix elements only. One step forward in this direction, starting from Gudkov matrices, from one side, and S-SDD matrices, from the other side, will be presented in this paper.

The Electronic Journal of Linear Algebra

In this article, some new results on $M$-matrices, $H$-matrices and their inverse classes are proved. Specifically, we study when a singular $Z$-matrix is an $M$-matrix, convex combinations of $H$-matrices, almost monotone $H$-matrices and Cholesky factorizations of $H$-matrices.

Linear Algebra and its Applications, 1998

We provide an algorithmic characterization of H-matrices. When A is an H-matrix, this algorithm determines a positive diagonal matrix D such that AD is strictly row diagonally dominant. In effect, D is produced iteratively by quantifying and * Work supported by a Natural Sciences and Engineering Research Council grant.

Linear Algebra and its Applications, 1997

We consider the class of doubly diagonally dominant matrices (A = [ ajj] E C", ', la,,1 l"jjl > Ck+ i laiklCk+ jlajkl. i #j)

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.