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

✍ Scribed by Jerzy K Baksalary; Oskar Maria Baksalary; Götz Trenkler

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 167 KB
• Volume: 372
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(03)00508-1

✦ Synopsis

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.