Expression for the multiplicative perturbation of the Moore-Penrose inverse

摘要

In this paper, we study the multiplicative perturbation of an operator of the form M = XNY, where X and Y are invertible. Based on this result, we study the Moore-Penrose invertibility of 2 x 2 block matrix M with Schur complement having closed range. We recover the cases when a Schur complement in M is either equal to zero or nonsingular. The necessary and sufficient conditions for the existence as well as the expressions for the Moore-Penrose inverses are obtained.