The perturbation of the Drazin inverse and oblique projection

K e y w o r d s I n d e x , Drazin inverse, Group inverse, Per turbat ion bound, Core rank. 1. I N T R O D U C T I O N A necessary and sufficient condition for the continuity of the Drazin inverse (to be defined in the next section) was established by Campbell and Meyer in 1975 [1]. They stated the main result: suppose tha t Aj, j = 1, 2 , . . . , and A are n × n matrices such tha t A j ~ A . Then A D -~ A D (where A D is the Drazin inverse of Aj) if and only if there is a positive integer J0 such tha t core r ankAj = core r ankA for j > j0 (where core r ankA = r ankA k, k = Ind(A), the index of A defined as the smallest integer k > 0 such tha t rank A k = rank Ak+l). In the same paper, they also indicated two difficulties in establishing norm est imates for the Drazin inverse. First, the Drazin inverse has a weaker type of "cancellation law" and is somewhat harder to work with algebraically than Moore-Penrose inverse. Also complicating things is the fact tha t the Jordan form is not a continuous function from C '~xn ~ C n x n and the Drazin inverse can be thought of in terms of the Jordan canonical form. Due to these reasons, they thought tha t it would be difficult to establish norm estimates for the Drazin inverse similar to those for the Moore-Penrose inverse, as was done by Stewart [2]. *Supported by National Nature Science Foundation of China, Doctoral Point Foundation of China, and Youth Science Foundation of Universities in Shanghai of China. tSuppor ted by the State Major Key Project for Basic Researches and the Doctoral Point Foundation of China. 0893-9659/00/$ see front mat ter (~ 2000 Elsevier Science Ltd. All rights reserved~ Typeset by A2~48.~ P Ih S0893-9659(99)00189-5