An improvement on the perturbation of the group inverse and oblique projection

The perturbations of the group inverse A(#) and oblique projection AA(#) of a square matrix A have been previously studied. Under certain assumptions on the matrix A and a perturbation matrix E, upper bounds for parallel toB(#)parallel to, parallel to BB(#)parallel to, parallel toB(#) - A(#)parallel to/parallel toA(#)parallel to and parallel to BB# - AA(#)parallel to/parallel to AA(#)parallel to, where B = A + E, are given in the literature. Recently, upper bounds for the general case have been published by Y Wei [Appl. Math. Comp. 98 (1999) 29]. However, the special cases in the literature and the continuity of the group inverse do not follow from the general upper bounds. In this paper, we derive new general upper bounds which not only cover all the special cases but also are sharper than Wei's results such that the continuity of the group inverse directly follows. A numerical example is given to illustrate the sharpness of the new general upper bounds. (C) 2001 Elsevier Science Inc. All rights reserved.