Special H-matrices and their Schur and diagonal-Schur complements

It is well known, see [D. Carlson, T. Markham, Schur complements of diagonally dominant matrices, Czech. Math. 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. showed that the similar statements hold for some special subclasses of H-matrices. The aim of this paper is to give more invariance results of this type, and simplified proofs for some already known results, by using scaling approach. The main idea of the considerations that follow is the fact that a matrix A is an H-matrix if and only if there exists a diagonal nonsingular matrix W such that AW is a strictly diagonally dominant (SDD) matrix. In other words, see [16], the class of H-matrices is diagonally derived from the class of SDD matrices. Some special subclasses of H-matrices could be characterized by the form of the corresponding scaling matrix W. These characterizations will be presented in short in the first section, as they have already been proven in [6], and some other subclasses of H-matrices will be recalled. In the second section simplified proofs of the statements from [14] will be presented, as well as another result of the same type concerning diagonal-Schur complement and Dashnic–Zusmanovich (DZ) matrices. The third section deals with another subclass of H-matrices, called S-Nekrasov matrices, for which we give some closure properties under taking the Schur complement and the diagonal -Schur complement in a similar way, i.e., by using scaling approach.