The generalized Schur complement in group inverses and (k + 1)-potent matrices

In this article, two facts related to the generalized Schur complement are studied. The first one is to find necessary and sufficient conditions to characterize when the group inverse of a partitioned matrix can be expressed in the Schur form. The other one is to develop a formula for any power of the generalized Schur complement of an idempotent partitioned matrix and then to characterize when this generalized Schur complement is a (k+1)-potent matrix. In addition, some spectral theory related to this complement is analyzed.