Additional results on index splittings for Drazin inverse solutions of singular linear systems
Yimin Wei, Hebing Wu · Electronic Journal of Linear Algebra · 2001
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(A k ) and N (U ) = N (A k ).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 A D f , 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 A D .The characteristics of the Drazin inverse solution A D f are established.Some criteria are given for comparing convergence rates ofMarek and Szyld on monotone-type iterations are extended.A characterization of the iteration matrix of an index splitting is also presented.