Majorization relations for Hadamard products
type: Item
instance of: scholarly article
title: Majorization relations for Hadamard products
zbMATH Open document ID: 0829.15010
DOI: 10.1016/0024-3795(94)00014-5
author: Yanyan Li
published in: Linear Algebra and its Applications
publication date: 1996-01-24T00:00:00Z
review text: \textit{C. R. Johnson} and \textit{R. B. Bapat} [Linear Algebra Appl. 104, 246- 247 (1988)] have conjectured that: if \(A\) and \(B\) are \(n \times n\) positive definite matrices with Hadamard product \(A \circ B\) then, for each \(k \leq n\), the product of the \(k\) smallest of the eigenvalues of \(A \circ B\) is at least as great as the product of the \(k\) smallest of the eigenvalues of \(AB\). This paper provides a proof of this conjecture and shows that the corresponding inequalities for the eigenvalues of \(A \circ B\) and \(AB^T\) and for the eigenvalues of \(A \circ B\) and the ``geometric mean'' \(A \# B : = A^{1/2} (A^{-1/2} BA^{-1/2})^{1/2} A^{1/2}\) are also true. \{Another proof of the Bapat-Johnson conjecture has been given by \textit{G. Visick}, ``A weak majorization involving the matrices \(A \circ B\) and \(AB\)'', Linear Algebra, Appl. 223-224, 731-744 (1995)\}.
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 778247
zbMATH Keywords: majorization
zbMATH Keywords: geometric mean
zbMATH Keywords: positive definite matrices
zbMATH Keywords: Hadamard product
zbMATH Keywords: eigenvalues
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)00014-5
OpenAlex ID: W2030793589
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description: scientific article; zbMATH DE number 778247