Rearrangement and matrix product inequalities
A relative of the rearrangement inequality of \textit{G. H. Hardy}, \textit{J. E. Littlewood} and \textit{G. Polya} [Inequalities (1934; Zbl 0010.10703); Chapter 6, A.3] is presented. Matrix product bounds are derived from there. Let \(\bar a_ i\), \(\bar b_ i\), \(i=1,2,...,n\), be two sets of real numbers. Denote by \(a_ i\), \(b_ i\), \(i=1,2,...,n\), the same two sets, when their components are arranged in the order: \(a_ 1\geq a_ 2\geq...\geq a_ n\), \(b_ 1\geq b_ 2\geq...\geq b_ n\). This notation applies to all sets of numbers, including eigenvalues and diagonal elements of matrices. The notation \(\lambda_ i(\cdot)\) is the ith eigenvalue of (\(\cdot)\). The following are the main theorems: Theorem 1. Let \(a_ i\), \(b_ i\) be two sets of real numbers with \(a_ i\geq 0\). Then for \(k=1,2,...,n\) one has \(\sum^{k}_{1}\bar a_ i\bar b_ i\leq \sum^{k}_{1}a_ ib_ i.\). Theorem 2. Let A, B be \(n\times n\) Hermitian matrices with \(A\geq 0\). Then for \(k=1,2,...,n\) one has \(\sum^{k}_{1}\lambda_{n-i+1}(AB)\leq \sum^{k}_{1}\lambda_ i(A)\lambda_ i(B).\) Theorem 3. Let A,B be \(n\times n\) Hermitian matrices with \(A\geq 0\). Then for \(k=1,2,...,n\) one has \(\sum^{k}_{1}\lambda_ i(A)\lambda_{n- i+1}(B)\leq \sum^{k}_{1}\lambda_ i(AB)\).
scientific article; zbMATH DE number 4173154