Maximizing traces of matrix functions
type: Item
instance of: scholarly article
title: Maximizing traces of matrix functions
zbMATH Open document ID: 1067.15014
published in: Linear Algebra and its Applications
publication date: 2004-08-04T00:00:00Z
review text: An infinitely differentiable function \(\varphi\colon I\rightarrow \mathbb{R}\) is said to be isoclinically metaconvex on \(I\) if whenever \(t_1,t_2\in I\), \(t_1\neq t_2\), and \(\varphi'(t_1)=\varphi'(t_2)\), then \(\varphi''(t_1)+\varphi''(t_2)>0\). If \(X\) and \(Y\) are two Hermitian \(n\times n\) matrices with eigenvalues \(x_1\geq x_2\geq\cdots\geq x_n\) and, respectively, \(y_1\geq y_2\geq\cdots\geq y_n\), it is proved that \[ \text{tr}(\varphi(X+Y))\leq \max_{\sigma\in S_n}\sum_{j=1}^n \varphi(x_j+y_{\sigma(j)}). \] Also, if \(A\) and \(B\) are positive definite \(n\times n\) matrices with eigenvalues \(a_1\geq a_2\geq \cdots \geq a_n>0\) and, respectively, \(b_1\geq b_2\geq \cdots \geq b_n>0\) and \(t\mapsto \varphi(\text{ e}^t)\) is isoclinically metaconvex, then it is proved that \[ \text{tr}(\phi(AB))\leq \max_{\sigma\in S_n}\sum_{j=1}^n \varphi(a_jb_{\sigma(j)}). \] An application to canonical correlations is also presented.
reviewed by: Aurelian Gheondea
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 2081246
zbMATH Keywords: eigenvalues
zbMATH Keywords: trace inequalities
zbMATH Keywords: canonical correlations
zbMATH Keywords: Hermitian matrix
zbMATH Keywords: positive definite matrix
author: S. W. Drury
MaRDI profile type: Publication
full work available at URL: j.laa.2004.02.029
OpenAlex ID: W2013694283
cites work: Q5690490
cites work: On symmetric functions of the eigenvalues of the sum of two Hermitian matrices
cites work: The canonical correlations of a 2\({\times}\)2 block matrix with given eigenvalues
cites work: Q3580354
cites work: Q3998992
cites work: Q4040931
cites work: Inequalities: theory of majorization and its applications
DOI: 10.1016/J.LAA.2004.02.029
description: scientific article; zbMATH DE number 2081246