Properties of Schur complements in partitioned idempotent matrices

Related to a complex partitioned matrix P, having A, B, C, and D as its consecutive m×m, m×n, n×m, and n×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 P1 and P2. We indicate conditions under which the idempotency of the sum P1+P2 and the difference P1−P2 is inherited by the sums and differences of the related Schur complements S1, S2 and T1, T2. 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.