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A Revisitation of Formulae for the Moore-Penrose Inverse of Modified Matrices
Linear Algebra and its Applications, (2003)
DOI: 10.1016/S0024-3795(03)00508-1
Аннотация
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.
