Generalized inverses of partitioned matrices in Banachiewicz–Schur form

Abstract

The problem of developing conditions under which generalized inverses of a partitioned matrix can be expressed in the so-called Banachiewicz–Schur form is reconsidered. Theorem of Marsaglia and Styan [Sankhyā Ser. A 36 (1974) 437], concerning the class of all generalized inverses, the class of reflexive generalized inverses, and the Moore–Penrose inverse, is strengthened and new results are established for the classes of outer inverses, least-squares generalized inverses, and minimum norm generalized inverses.