Inequalities on the singular values of an off-diagonal block of a Hermitian matrix
type: Item
instance of: scholarly article
title: Inequalities on the singular values of an off-diagonal block of a Hermitian matrix
zbMATH Open document ID: 0958.15017
DOI: 10.1155/S1025583499000090
author: Chi-Kwong Li
author: Roy Mathias
published in: Journal of Inequalities and Applications
publication date: 2001-04-09T00:00:00Z
full work available at URL: https://eudml.org/doc/120868
review text: Given two real vectors \(x\), \(y\) in \(\mathbb{R}^n\), we say that \(x\) is weakly majorized by \(y\) if the sum of the \(k\) largest entries of \(x\) is not larger than that of \(y\) for each \(k= 1,\dots, n\). Let \(\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_n\) denote the eigenvalues of an \(n\times n\) Hermitian matrix \(H\). For an \(m\times n\) complex matrix \(A\), let \(\sigma_i(A)(=\sqrt{\lambda_i(A^*A)})\) denote the \(i\)th singular value of \(A\) for \(i= 1,\dots, k\), where \(k= \min\{m, n\}\), and let \(\sigma= (\sigma_1,\dots, \sigma_k)\) be the vector of singular values of \(A\). The authors give two proofs of the following theorem. Let \(H\) be an \(n\times n\) Hermitian matrix. Then for any unitary matrix \(U\) of the form \([X|Y]\), where \(X\) is an \(n\times k\) matrix, the vector \(\sigma(X^* HY)\) is majorized by the vector \({1\over 2}(\lambda_1- \lambda_n,\dots, \lambda_m- \lambda_{n-m+1})\), where \(m= \min\{k, n-k\}\). Consequently, for any Schur convex increasing function \(f: \mathbb{R}^m\to \mathbb{R}\), we have \[ f(\sigma(X^* HY))\leq f(\textstyle{{1\over 2}} (\lambda_1- \lambda_n,\dots, \lambda_m- \lambda_{n- m+1})). \] If we take \(f(x)= \sum^m_{j= 1}x^2_i\) in this theorem, we obtain a result that was conjectured by J. Durbin, and proved independently in a paper by \textit{P. Bloomfield} and \textit{G. S. Watson} [Biometrika 62, 121-128 (1975; Zbl 0308.62056)] and by \textit{M. Knott} [ibid. 62, 129-132 (1975; Zbl 0308.62057)], and later in a paper by \textit{Z. Jia} [An extension of Styan's inequality (in Chinese), Gongcheng Shuxue Xuebao (J. Eng. Math.) 13, 122-126 (1996)]. In a note added proof, the author states that ``X. Zhan has another proof of our main theorem, which was also obtained independently by R. Bhatia, F. C. Silva, P. Assouad and J. A. Dias da Silva''.
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 1287992
zbMATH Keywords: inequalities involving singular values
zbMATH Keywords: eigenvalues
zbMATH Keywords: Hermitian matrix
zbMATH Keywords: singular value
reviewed by: M. Zuhair Nashed
MaRDI profile type: Publication
OpenAlex ID: W2018254026
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description: scientific article; zbMATH DE number 1287992