A revisitation of formulae for the Moore–Penrose inverse of modified matrices

Formulae for the Moore–Penrose inverse M+ of rank-one-modifications of a given m×n complex matrix A to the matrix M=A+bc*, where b and c* are nonzero m×1 and 1×n complex vectors, are revisited. An alternative to the list of such formulae, given by Meyer [SIAM J. Appl. Math. 24 (1973) 315] in forms of subtraction–addition type modifications of A+, is established with the emphasis laid on achieving versions which have universal validity and are in a strict correspondence to characteristics of the relationships between the ranks of M and A. Moreover, possibilities of expressing M+ as multiplication type modifications of A+, with multipliers required to be projectors, are explored. In the particular case, where A is nonsingular and the modification of A to M reduces the rank by 1, such a possibility was pointed out by Trenkler [R.D.H. Heijmans, D.S.G. Pollock, A. Satorra (Eds.), Innovations in Multivariate Statistical Analysis. A Festschrift for Heinz Neudecker, Kluwer, London, 2000, p. 67]. Some applications of the results obtained to various branches of mathematics are also discussed.