Block idempotent matrices and generalized Schur complement

In this paper, we study the idempotency of the generalized Schur complement on generalized inverse A T , S ( 2 ) , together with other familiar generalized inverses such as A M , N † , A † , A d and A g , defined from a partitioned idempotent matrix, which is a complementarity of [Jerzy K. Baksalary, Oskar Maria Baksalary, Tomasz Szulc, Properties of Schur complements in partitioned idempotent matrices, Lin. Alg. Appl. 379 (2004) 303–318; D.V. Ouellette, Schur complememnts and statistics, Lin. Alg. Appl. 36 (1981) 187–295].