Some inequalities on generalized Schur complements (Q1970437)

Let \(A=[A_{ij}]\) denote a block matrix of order two with square diagonal blocks. The generalized Schur complement \(S_1(A)\) of \(A_{11}\) is defined by \(S_1(A)=A_{22} -A_{21}A^+_{11} A_{12}\) where \(A^+_{11}\) denotes the Moore-Penrose pseudoinverse of \(A_{11}\) so that \(S_1(A)\) is defined also for singular matrices. For a Hermitian matrix \(A\), the authors prove some new inequalities that involve the vectors of the eigenvalues of \(S_1(A)^+\), \(A^+_{22},A_{22}\) and \(S_1(A^k)\). Moreover, they give conditions on \(C\) such that \(S_1(C^* AC)\leq S_1(C^*)A_{22} S_1(C)\) in the ordering defined by the positive definiteness.