A minimum principle and estimates of the eigenvalues for Schur complements of positive semidefinite Hermitian matrices
Let \(A\) be a positive definite \(n\times n\) matrix. For any subsets \(\alpha \) and \(\beta\) of the indices, \(A(\alpha,\beta)\) denotes the corresponding submatrix of \(A\), and \(A(\alpha):= A(\alpha,\beta)\). For each subset \(\alpha \) there is a generalized Schur complement \(A/A(\alpha)\) of \(A\) defined by \(A(\alpha') - A(\alpha',\alpha)A(\alpha)^{-1}A(\alpha,\alpha')\) where \(\alpha' \) is the complement of \(\alpha \). \textit{R. L. Smith} [Linear Algebra Appl. 177, 137-144 (1992; Zbl 0765.15007)] has proved an interlacing theorem for the eigenvalues of \(A/A(\alpha)\) and \(A\). The authors of the present paper prove a series of other inequalities involving the eigenvalues of \(A/A(\alpha)\) which are analogous to known results on eigenvalues of sums and products of positive definite matrices. Most of these are too complicated to state here, but a simple example is given by the following (a Remark to Theorem 5). Let \(A\) and \(B\) be positive definite \(n\times n\) matrices. Then \[ \biggl({\det(A+B)\over\det(A+B)(\alpha)}\biggr)^{1/k}\geq \biggl({\det(A)\over\det(A(\alpha))}\biggr)^{1/k}+\biggl({\det(B)\over\det(B(\alpha))}\biggr)^{1/k} \] for any \(\alpha\subseteq\{1,\ldots,n\}\) of size \(n-k\).
scientific article; zbMATH DE number 1076374
scientific article