Some inequalities for eigenvalues of Schur complements of Hermitian matrices (Q2503016)

Let \(\mathbb{C}^{m\times n}\) denote the set of \(m\times n\) complex matrices. Let \(A\in \mathbb{C}^{n\times n}.\) The Moore-Penrose inverse of \(A,\) denoted by \(A^\dag ,\) is the unique solution of the matrix equations \(AXA=A, XAX=X, (AX)^*=AX, (XA)^*=XA.\) Let \(N=\{1,2,\dots ,n\}.\) For nonempty index sets \(\alpha, \beta \subset N,\) let \(A(\alpha, \beta )\) denote the submatrix of \(A\in \mathbb{C}^{n\times n}\) lying in the rows indicated by \(\alpha \) and the columns indicated by \(\beta .\) Let \(\alpha \subset N\) and \(\alpha ^c=N-\alpha ,\) both arranged in inreasing order. Then \(A/\alpha =A(\alpha ^c,\alpha ^c)-A(\alpha ^c,\alpha )[A(\alpha ,\alpha )]^\dag A(\alpha ,\alpha ^c)\) is called the generalized Schur complement of \(A\) with respect to \(A(\alpha ,\alpha ).\) Let \(B\in \mathbb{C}^{m\times n}\) and \(A\) be positive semidefinite or Hermitian. The authors prove several inequalities for the eigenvalues of the generalized Schur complement of the matrix \(BAB^*\) in terms of the eigenvalues of the generalized Schur complements of \(BB^*\) and \(A.\)