Some inequalities on generalized Schur complements

AbstractThis paper presents some inequalities on generalized Schur complements. Let A be an n×n (Hermitian) positive semidefinite matrix. Denote by A/α the generalized Schur complement of a principal submatrix indexed by a set α in A. Let A+ be the Moore–Penrose inverse of A and λ(A) be the eigenvalue vector of A. The main results of this paper are: 1.λ(A+(α′))⩾λ((A/α)+), where α′ is the complement of α in {1,2,…,n}.2.λ(Ar/α)⩽λr(A/α) for any real number r⩾1.3.(C*AC)/α⩽C*/αA(α′)C/α for any matrix C of certain properties on partitioning.