An invariance property related to the reverse order law

✍ Scribed by Jerzy K. Baksalary; Oskar Maria Baksalary

• Publisher: Elsevier Science
• Year: 2005
• Tongue: English
• Weight: 96 KB
• Volume: 410
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2005.08.006

✦ Synopsis

It is shown that, quite surprisingly, all matrices of the form L -M -, where L -and M - denote generalized inverses of L and M, are generalized inverses of ML if and only if the product MLL -M -ML is invariant with respect to the choice of L -and M -, which at the first glance looks to be a weaker condition than the requirement that MLL -M -ML = ML for every L -and M -. This statement follows as an immediate corollary to the main result of the present note, which provides two criteria for the invariance of expressions of the type KL -M -N involving four given matrices K, L, M, N, with generalized inverses L -, M -of two of them.