Generalized inverses of partitioned matrices and recalculation of least squares estimates for data or model changes

Abstract

Let A be a m × n matrix and a be an m × 1 vector. In this paper, the authors have obtained formulae for computing several types of g-inverses of (A: α) from the corresponding g-inverses of A. Similar formulae are obtained for computing g-inverses of A from those of (A: α). These formulae are useful for revising least squares estimates in the following situations: (i) deletion or addition of an observation; and (ii) deletion or addition of a parameter in the linear model. Finally, the paper also gives formulae for computing g-inverses of A+BDC* from those of A when D is nonsingular.