Some generalized inverses of partition matrix and quotient identity of generalized Schur complement

In this paper, we first define some generalized Schur complements, then we study the representation of M–P inverse M1† group inverse M1g and Drazin M1d for the partitioned matrix M1=ABCD. Based on this, we give three kind quotient identities of generalized Schur complement about the partitioned matrix K=ABECDFGHL and develops the results in paper [M. Redivo-Zaglia, Pseudo-Schur complement and their applications, Appl. Num. Math. 50 (2004) 511–519; K. Jbilou, A. Messaoudi, K. Tabaâ, Some Schur complement identities and applications to matrix exatrapolation methods, Linear Algebra Appl. 392 (2004) 195–210]. In the end of the paper, we give the application of these generalized Schur complement in solution of linear equation, and give numerical example to show our results.