Eigenvalue equalities for ordinary and Hadamard products of powers of positive semidefinite matrices
type: Item
instance of: scholarly article
title: Eigenvalue equalities for ordinary and Hadamard products of powers of positive semidefinite matrices
zbMATH Open document ID: 1123.15004
author: Che-Man Cheng
author: Ieng-Chi Law
author: Sok-I Leong
published in: Linear Algebra and its Applications
publication date: 2007-04-10T00:00:00Z
review text: Let \(A\) and \(B\) be \(n\times n\) positive semidefinite matrices and \(0<\alpha <\beta.\) Let \(A\circ B\) denote the Hadamard product of \(A\) and \(B,\) and \([A]_l\) denote the leading \(l\times l\) principle submatrix of \(A.\) Let \(\lambda_1(X)\geq \cdots \geq \lambda_n(X)\) denote the eigenvalues of an \(n\times n\) matrix \(X\) ordered when they are all real. In this paper, those matrices that satisfy any of the equalities \[ \lambda _i^{1/\alpha }(A^\alpha B^\alpha)=\lambda _i^{1/\beta }(A^\beta B^\beta),\quad i=1,n \] \[ \lambda _i^{1/\alpha }([A^\alpha ]_l)=\lambda _i^{1/\beta }([A^\beta ]_l ),\quad i=1,\dots ,l \] \[ \lambda _i^{1/\alpha }(A^\alpha \circ B^\alpha)=\lambda _i^{1/\beta }(A^\beta \circ B^\beta),~~i=1,\dots ,n \] are determined. Several related results are discussed.
reviewed by: Jaspal Singh Aujla
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 5141650
zbMATH Keywords: eigenvalues
zbMATH Keywords: Hadamard product
zbMATH Keywords: positive semidefinite matrix
MaRDI profile type: Publication
full work available at URL: j.laa.2006.12.007
OpenAlex ID: W2129126720
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DOI: 10.1016/J.LAA.2006.12.007
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description: scientific article; zbMATH DE number 5141650