Further inequalities involving the Khatri-Rao product
type: Item
instance of: scholarly article
title: Further inequalities involving the Khatri-Rao product
zbMATH Open document ID: 1179.15022
author: Shuangzhe Liu
author: Götz Trenkler
author: Zhongpeng Yang
published in: Linear Algebra and its Applications
publication date: 2009-05-12T00:00:00Z
review text: Inequalities involving Khatri-Rao and Tracy-Singh matrix products, which are extension of the Hadamard and Kronecker products to the case of partitioned matrices, respectively, are studied. Results, previously presented by \textit{G. P. H. Styan} [Linear Algebra Appl. 6, 217--240 (1973; Zbl 0255.15002)] on companion inequalities involving the Hadamard product using statistical reasoning in the context of multivariety analysis, are extended. New results including companion inequalities are established and the conditions for such inequalities to become equalities are given. Especially, new results based on the Schur complement method are presented in the case of non-singular Hermitian matrices where no positive definite matrices are required. In the positive definite matrix case, the Kantorovich-type inequalities are established.
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A69
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 5552968
zbMATH Keywords: Hadamard product
zbMATH Keywords: Khatri-Rao product
zbMATH Keywords: Tracy-Singh product
zbMATH Keywords: Kronecker product
zbMATH Keywords: Schur complement
zbMATH Keywords: matrix inequalities
zbMATH Keywords: non-singular Hermitian matrices
zbMATH Keywords: positive definite matrix
zbMATH Keywords: Kantorovich-type inequalities
MaRDI profile type: Publication
full work available at URL: j.laa.2008.12.004
OpenAlex ID: W1976790778
cites work: Concavity of certain maps on positive definite matrices and applications to Hadamard products
cites work: The Schur complement and its applications
cites work: Q5568891
cites work: Block Kronecker products and the vecb operator
cites work: Inequalities involving Hadamard products of positive semidefinite matrices
cites work: Several inequalities involving Khatri-Rao products of positive semidefinite matrices
cites work: Equality conditions for matrix Kantorovich-type inequalities
cites work: Q3527509
cites work: Hadamard products and multivariate statistical analysis
cites work: A new matrix product and its applications in partitioned matrix differentiation
cites work: A quantitative version of the observation that the Hadamard product is a principal submatrix of the Kronecker product
cites work: Q4549920
cites work: Schur complements and matrix inequalities in the Löwner ordering
DOI: 10.1016/J.LAA.2008.12.004
description: scientific article; zbMATH DE number 5552968