A minimum principle and estimates of the eigenvalues for Schur complements of positive semidefinite Hermitian matrices
type: Item
instance of: scholarly article
title: A minimum principle and estimates of the eigenvalues for Schur complements of positive semidefinite Hermitian matrices
zbMATH Open document ID: 0885.15010
DOI: 10.1016/S0024-3795(96)00595-2
author: Jianzhou Liu
author: L. Zhu
published in: Linear Algebra and its Applications
publication date: 1998-04-08T00:00:00Z
review text: 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 \(n-k\).
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 1076374
zbMATH Keywords: inequalities for determinants
zbMATH Keywords: Schur complement
zbMATH Keywords: eigenvalues
zbMATH Keywords: inequalities
zbMATH Keywords: positive definite matrices
reviewed by: John D. Dixon
MaRDI profile type: Publication
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cites work: Q4040931
cites work: Some interlacing properties of the Schur complement of a Hermitian matrix
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description: scientific article; zbMATH DE number 1076374