Kantorovich and Cauchy-Schwarz inequalities involving positive semidefinite matrices, and efficiency comparisons for a singular linear model
type: Item
instance of: scholarly article
title: Kantorovich and Cauchy-Schwarz inequalities involving positive semidefinite matrices, and efficiency comparisons for a singular linear model
zbMATH Open document ID: 0881.15020
DOI: 10.1016/S0024-3795(96)00284-4
author: Shuangzhe Liu
author: Heinz Neudecker
published in: Linear Algebra and its Applications
publication date: 1998-02-22T00:00:00Z
review text: The authors prove that \(Y^+UY^{+'}\leq{(\lambda_1+\lambda_r)^2\over 4\lambda_1\lambda_r}(Y'U^+ Y)\), where the \(n\times n\) positive semidefinite matrix \(U\geq 0\) of rank \(r\) has nonzero eigenvalues \(\lambda_1,\dots,\lambda_r\), \(Y\) is of order \(n\times k\), and \(R(Y)\subset R(U)\); \((\cdot)^+\) indicates the Moore-Penrose inverse and \(R(\cdot)\) indicates the column space of a matrix. This result is generalized for two positive semi-definite matrices. Several Cauchy-Schwarz-type inequalities are briefly studied. Some of these results are used to compare several efficient and inefficient estimators for a singular linear model.
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 62J05
zbMATH DE Number: 1044234
zbMATH Keywords: matrix Kantorovich inequality
zbMATH Keywords: positive semidefinite matrix
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: Cauchy-Schwarz-type inequalities
zbMATH Keywords: singular linear model
reviewed by: Mihail Voicu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1044234