Extremal Characterizations of the Schur Complement and Resulting Inequalities

Let H = ( H11 H12 H∗ 12 H22 ) be an n×n positive semidefinite matrix, where H11 is k×k with 1 ≤ k < n. The generalized Schur complement of H11 in H is defined as S(H) = H22 −H∗ 12H † 11H12, where H 11 is the Moore-Penrose generalized inverse of H11. It has the extremal characterizations S(H) = max { W : H − ( 0k 0 0 W ) ≥ 0,W is (n− k)× (n− k) Hermitian } and S(H) = min {[Z|In−k]H[Z|In−k] : Z is (n− k)× k} . These characterizations are used to deduce many old and new inequalities for Schur complements of positive semidefinite matrices. In many cases, stronger statements and shorter proofs can be obtained using the extremal characterizations. ∗Both authors were supported by NSF grants DMS-9504795 and DMS-9704534.