Some inequalities on generalized Schur complements

This paper presents some inequalities on generalized Schur complements. Let A be an n x n (Hermitian) positive semidefinite matrix. Denote by A/alpha the generalized Schur complement of a principal submatrix indexed by a set a in A. Let A(+) be the Moore-Penrose inverse of A and lambda(A) be the eig envalue vector of A. The main results of this paper are: 1 lambda(A(+)(alpha')) greater than or equal to lambda((A/alpha)(+)), where alpha' is the complement of alpha in {1,2,..., n}. 2. lambda(A(r)/alpha) less than or equal to lambda(r)(A/alpha) for any real number r greater than or equal to 1. 3. (C*AC)/alpha less than or equal to C*/alpha A(alpha') C/alpha for any ma trix C of certain properties on partitioning. (C) 1999 Elsevier Science Inc . All rights reserved.