Related to a complex partitioned matrix P, having A, B, C, and D as its consecutive m Γ m, m Γ n, n Γ m, and n Γ n submatrices, are generalized Schur complements S = A -BD -C and T = D -CA -B, where the minus superscript denotes a generalized inverse of a given matrix. In the first part of the present paper, we aim at specifying conditions under which certain properties of P hold also for S and T when P is an idempotent matrix (i.e., represents a projector) or a Hermitian idempotent matrix (i.e., represents an orthogonal projector). Among the properties considered are: the idempotency itself, existence of an eigenvalue equal to zero, and relationships between eigenvectors of P and those of S and T, corresponding to this eigenvalue. The second part of the paper deals with two partitioned idempotent matrices P 1 and P 2 . We indicate conditions under which the idempotency of the sum P 1 + P 2 and the difference P 1 -P 2 is inherited by the sums and differences of the related Schur complements S 1 , S 2 and T 1 , T 2 . The inheritance property of such a type is also discussed in the context of matrix partial orderings, with the emphasis laid on the minus (rank subtractivity) ordering.