A direct proof and a generalization for a Kantorovich type inequality
type: Item
instance of: scholarly article
title: A direct proof and a generalization for a Kantorovich type inequality
zbMATH Open document ID: 1074.15025
author: Jieyong Zhou
author: Jianguo Huang
published in: Linear Algebra and its Applications
publication date: 2005-05-04T00:00:00Z
review text: The following Kantorovich type inequality has important applications in statistics; it was proved by \textit{F. L. Bauer} and \textit{A. S. Householder} [Numer. Math. 2, 308--311 (1960; Zbl 0104.34502)]. Let \(A\) be a real symmetric positive definite \(n\times n\) matrix with eigenvalues \(0< \lambda_n\leq\cdots\leq\lambda_1\), and let \(x\), \(y\) be nonzero real \(n\)-columns. Suppose \({|x^T y|\over\| x\|_2\| y\|_2}\geq \cos(\theta)\) for some \(0\leq\theta\leq{\pi\over 2}\), and define \(\kappa= ({\lambda_1\over\lambda_n}){1+ \sin(\theta)\over 1-\sin(\theta)}\). Then \({(x^Ty)^2\over (y^T A^{-1}y)(x^T Ax)}\geq {4\over \kappa+ 2+ 1/\kappa}\). The original proof was technical and lengthy, but a new simple proof is presented here, which makes use of an auxiliary optimization problem. A generalization to a matrix form of the inequality is also established, by using compound matrices.
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 2165355
zbMATH Keywords: positive definite matrix
zbMATH Keywords: Kantorovich type inequality
zbMATH Keywords: eigenvalues
MaRDI profile type: Publication
full work available at URL: j.laa.2004.10.009
OpenAlex ID: W1976694838
cites work: Some inequalities involving the euclidean condition of a matrix
cites work: Q4226179
cites work: Interior Methods for Nonlinear Optimization
cites work: Inexact Preconditioned Conjugate Gradient Method with Inner-Outer Iteration
cites work: Matrix Analysis
cites work: Q5543004
cites work: Q4016506
cites work: Some further matrix extensions of the Cauchy-Schwarz and Kantorovich inequalities, with some statistical applications
cites work: Q3998482
cites work: A matrix version of the Wielandt inequality and its applications to statistics
DOI: 10.1016/J.LAA.2004.10.009
description: scientific article; zbMATH DE number 2165355