Block idempotent matrices and generalized Schur complement

In this paper, we study the idempotency of the generalized Schur complement on generalized inverse AT,S(2), together with other familiar generalized inverses such as AM,N†, A†, Ad and Ag, 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].