Relationships between generalized inverses of a matrix and generalized inverses of its rank-one-modifications

✍ Scribed by Jerzy K Baksalary; Oskar Maria Baksalary

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 215 KB
• Volume: 388
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2004.03.020

✦ Synopsis

For given complex matrix A and nonzero complex vectors b, c, relationships between generalized inverses of A and generalized inverses of the rank-one-modified matrix M = A + bc * (with c * being the conjugate transpose of c) are investigated. The following three questions are considered: (i) when a given generalized inverse A -belongs to the set M{1} of all generalized inverses of M, (ii) when does A -∈ A{1} exist such that simultaneously A -∈ M{1}, and (iii) when the set A{1} is a subset of M{1}. The same questions are also discussed for reflexive generalized inverses of A and M. The answers obtained are commented from the view-point of a result concerning comparison of ranks of M and A.