A Generalization of the Schur Complement by Means of the Moore–Penrose Inverse

Suppose the complex matrix M is partitioned into a 2 × 2 array of blocks; let M₁₁ = A,M₁₂ = B,M₂₁ = C,M₂₂ = D. The generalized Schur complement of A in M is defined to be M/A = D - CA^ + B, where A^ + is the Moore–Penrose inverse of A. The relationship of the ranks of M, A, and $M/A$ is determined. A new proof, under certain conditions, of Sylvester’s determinantal formula is given. A quotient formula like one previously proved for the Schur complement is obtained. Finally, several known inequalities for positive semidefinite Hermitian matrices are generalized.