Further results on H-matrices and their Schur complements

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. (c) 2007 Elsevier Inc. All rights reserved.