Generalizations of Diagonal Dominance in Matrix Theory

A matrix A ∈ C is called generalized diagonally dominant or, more commonly, an H−matrix if there is a positive vector x = (x1, · · · , xn) such that |aii|xi > ∑ j 6=i |aij|xj, i = 1, 2, · · · , n. In this thesis, we first give an efficient iterative algorithm to calculate the vector x for a given H-matrix, and show that this algorithm can be used effectively as a criterion for H-matrices. When A is an H-matrix, this algorithm determines a positive diagonal matrix D such that AD is strictly (row) diagonally dominant; its failure to produce such a matrix D signifies that A is not an H-matrix. Subsequently, we consider the class of doubly diagonally dominant matrices (abbreviated d.d.d.). We give necessary and sufficient conditions for a d.d.d. matrix to be an H-matrix. We show that the Schur complements of a d.d.d matrix are also d.d.d. matrices, which can be viewed as a natural extension of the corresponding result on diagonally dominant matrices. Lastly, we obtain some results on the numerical stability of incomplete block LU factorizations of H-matrices and answer a question posed in the literature.