Additional results on index splittings for Drazin inverse solutions of singular linear systems

Given an n × n singular matrix A with Ind(A) = k, an index splitting of A is one of the form A = U − V , where R(U) = R(Ak) and N(U) = N(Ak). This splitting, introduced by the first author, generalizes the proper splitting proposed by Berman and Plemmons. Regarding singular systems Au = f , the first author has shown that the iterations u(i+1) = U#V u(i) + U#f converge to ADf , the Drazin inverse solution to the system, if and only if the spectral radius of U#V is less than one. The aim of this paper is to further study index splittings in order to extend some previous results by replacing the Moore-Penrose inverse A+ and A−1 with the Drazin inverse AD. The characteristics of the Drazin inverse solution ADf are established. Some criteria are given for comparing convergence rates of U i Vi, where A = U1 − V1 = U2 − V2. Results of Collatz, Marek and Szyld on monotone-type iterations are extended. A characterization of the iteration matrix of an index splitting is also presented.