Inequalities on the singular values of an off-diagonal block of a Hermitian matrix
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''.
scientific article; zbMATH DE number 1287992
scientific article