Multiplicative perturbation theory of the Moore-Penrose inverse and the least squares problem

Abstract

Bounds for the variation of the Moore Penrose inverse of general matrices under multiplicative perturbations are presented. Their advantages with respect to classical bounds under additive perturbations and with respect to other bounds under multiplicative perturbations available in the literature are carefully studied and established. Closely connected to these developments a complete multiplicative perturbation theory for least squares problems, valid for perturbations of any size, is also presented, improving in this way recent multiplicative perturbation bounds which are valid only to first order in the size of the perturbations. The results in this paper are mainly based on exact expressions of the perturbed Moore-Penrose inverse in terms of the unperturbed one, the perturbation matrices, and certain orthogonal projectors. Such feature makes the new results amenable to be generalized in the future to linear operators in infinite dimensional spaces.