On majorization and Schur products
Let \(A\) be a Hermitian matrix, \(D_1,\ldots, D_m\) square matrices and \(X=\sum D_kAD^*_ k\). This paper concerns the relationship between the spectra of \(A\) and \(X\). It is shown that if \(\sum D_kD^*_ k=\sum D^*_kD_k = I\), then the spectrum of \(X\) is majorized by that of \(A\). For the general case a result in terms of weak majorization is obtained. If the \(D_k\) are diagonal matrices, then \(X\) is the Schur (or Hadamard) product of \(A\) and a positive semidefinite matrix, thus the results are applicable to spectra of Schur products of positive semidefinite matrices. Several special cases are considered and a stronger version of a conjecture of Marshall and Olkin is proved.
scientific article; zbMATH DE number 3922881