A generalization of irreducibility and diagonal dominance with applications to horizontal and vertical linear complementarity problems

Abstract

In this paper, we generalize and analyze the concepts of diagonal dominance and irreducibility in the framework of column and row representative matrices of a set. Our analysis includes the definition of particular sets of M- and H-matrices. We also analyze the form that the matrices of the introduced irreducible sets must have and the implications of the obtained results on the solution of vertical and horizontal linear complementarity problems. In this context, we prove that the projected Jacobi and the projected Gauss-Seidel methods for horizontal linear complementarity problems converge when the matrices of the problem satisfy one of the introduced generalizations of strict diagonal dominance or of irreducible diagonal dominance. This extends current convergence results.