On majorization and Schur products
type: Item
instance of: scholarly article
title: On majorization and Schur products
zbMATH Open document ID: 0577.15016
DOI: 10.1016/0024-3795(85)90147-8
author: V. S. Sunder
author: R. B. Bapat
published in: Linear Algebra and its Applications
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 3922881
zbMATH Keywords: Schur product
zbMATH Keywords: Hadamard product
zbMATH Keywords: Hermitian matrix
zbMATH Keywords: spectrum
zbMATH Keywords: weak majorization
zbMATH Keywords: positive semidefinite matrices
reviewed by: Rabe von Randow
MaRDI profile type: Publication
full work available at URL: 0024-3795(85)90147-8
OpenAlex ID: W2039859942
Wikidata QID: Q99283318
cites work: Q5831338
cites work: Inequalities: theory of majorization and its applications
cites work: On permutations, convex hulls, and normal operators
description: scientific article; zbMATH DE number 3922881