A survey of Cauchy-Schwarz and Kantorovich-type matrix inequalities
type: Item
instance of: scholarly article
title: A survey of Cauchy-Schwarz and Kantorovich-type matrix inequalities
zbMATH Open document ID: 0915.62056
DOI: 10.1007/BF02927110
author: Shuangzhe Liu
author: Heinz Neudecker
published in: Statistical Papers
publication date: 1999-05-18T00:00:00Z
review text: Vector and matrix inequalities play important rules in various mathematical and statistical contexts. The number of results and applications grows day by day. Our purpose is to survey Cauchy-Schwarz and Kantorovich-type vector and matrix inequalities, to present related matrix results, and to give examples of applications of Cauchy-Schwarz and Kantorovich-type matrix inequalities in statistics and econometrics. The structure is as follows. Section 2 collects Cauchy-Schwarz and Kantorovich-type vector inequalities, which have been developed and applied widely. For this (vector) case we omit almost all proofs. Section 3 contains three useful lemmas. We apply them in Section 4 to derive Cauchy-Schwarz and Kantorovich-type matrix inequalities. These are the main results of the paper. Section 5 contains additional matrix inequalities which can be viewed as applications of the main results to sums of matrices, (upper left) submatrices and other matrix expressions. The last section contains applications in statistics and econometrics. All matrices, vectors and scalars considered throughout the paper will be real.
Mathematics Subject Classification ID: 62H99
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 62P20
zbMATH DE Number: 1251499
zbMATH Keywords: Cauchy-Schwarz inequalities
zbMATH Keywords: Kantorovich inequalities
MaRDI profile type: Publication
cites work: Q4347918
cites work: Generalized matrix versions of the Cauchy-Schwarz and Kantorovich inequalities
cites work: The inefficiency of least squares
cites work: Q3803901
cites work: Some matrix inequalities
cites work: Comparison of the Variance of Minimum Variance and Weighted Least Squares Regression Coefficients
cites work: Q5791786
cites work: Some extensions of the Kantorovich inequality and statistical applications
cites work: Q3711519
cites work: Q5813488
cites work: Bounds for the trace of the difference of the covariance matrices of the OLSE and BLUE
cites work: Kantorovich inequalities and efficiency comparisons for several classes of estimators in linear models
cites work: Several matrix Kantorovich-type inequalities
cites work: Kantorovich and Cauchy-Schwarz inequalities involving positive semidefinite matrices, and efficiency comparisons for a singular linear model
cites work: Comparison of Least Squares and Minimum Variance Estimates of Regression Parameters
cites work: Inequalities: theory of majorization and its applications
cites work: Matrix versions of the Cauchy and Kantorovich inequalities
cites work: A matrix inequality including that of Kantorovich
cites work: Q4285678
cites work: A matrix version of the Ky Fan generalization of the kantorovich inequality
cites work: Some further matrix extensions of the Cauchy-Schwarz and Kantorovich inequalities, with some statistical applications
cites work: Q3326646
cites work: The inefficiency of least squares: Extensions of the Kantorovich inequality
cites work: Differences of means
cites work: Some comments on six inequalities associated with the inefficiency of ordinary least squares with one regressor
full work available at URL: https://doi.org/10.1007/bf02927110
OpenAlex ID: W2068945207
description: scientific article; zbMATH DE number 1251499