Diagonal scaling of a special type and its benefits

Abstract

For certain subclasses of H-matrices we will exploit the structure of their scaling matrix to improve known results in several areas of applied linear algebra: convergence theory of relaxation methods, estimation of infinity norm of the matrix inverse and determinant estimation. Block vs partitioned H-matricesIn the literature concerning generalizations of diagonal dominance suited for large matrices, one can find two different, but similar, approaches. One is referred to as block H-matrices (see for example [3]), and the other one recently introducedpartitioned H-matrices, [1,2]. The aim of this report is to investigate the relationship between these two approaches and show possible benefits that can be obtained by combining them.We start with basic definitions. A matrix A = [a ij ] ∈ C n,n is called:• M-matrix if it is a real matrix of Z-form (for all different indices i and j, a ii 0 and a ij 0), A is nonsingular, and A −1 0.• H-matrix (i.e. generalized diagonally dominant, GDD), if its comparison matrix A = [m ij ], defined as follows,Two well known properties that are the foundation for generalisation of diagonal dominance via partitions are the following.• A real matrix with Z-form, is an M-matrix if and only if all of its principal minors are positive.• A complex matrix A = [a ij ] ∈ C n,n is GDD, i.e. H, if and only if AX is SDD, for someDenoting the family of all partitions of the index set {1, 2, ..., n} into m disjoint, nonempty proper subsets byfor a given partition S ∈ P(n, m) a matrix A = [a ij ] ∈ C n,n is called P H(S) if and only if A i is an M-matrix, ∀i ∈ S, where A i = [d ij ] (called aggregated matrix) is given by:The other approach mentioned above is also via partitions but yet in a different way. Namely, given a partition J ∈ P(n, N ) we define its corresponding block comparison matrix A J = [c ij ] as:∞ , c ij = −||A ij || ∞ , i, j = 1, 2,