Moore-Penrose inverse of sums of operators and matrices

Abstract

Under certain conditions, we prove that the Moore-Penrose inverse of a sum of operators is the sum of the Moore-Penrose inverses. From this, we derive characterizations and expressions for the Moore-Penrose inverse of an operator that are useful for its computation. We give formulations of them for finite matrices. We show that results appeared in the literature, and which were proved with different approaches, can be derived as particular cases of some of these formulations. We present an application to distance matrices of weighted trees.