Norm inequalities for Cartesian decompositions
type: Item
instance of: scholarly article
title: Norm inequalities for Cartesian decompositions
zbMATH Open document ID: 0962.15017
DOI: 10.1016/S0024-3795(98)10174-X
author: Xingzhi Zhan
published in: Linear Algebra and its Applications
publication date: 2001-06-01T00:00:00Z
review text: Let \(T=A+iB\) be the Cartesian decomposition of a complex matrix \(T\) with \(A,B\) Hermitian. Let \(\alpha_j\), \(\beta_j\) be the eigenvalues of \(A\) and \(B\), respectively, ordered such that \(|\alpha_1|\geqslant \dots \geqslant |\alpha_n|\) and \(|\beta_1|\geqslant \dots \geqslant |\beta_n|\). \textit{T. Ando} and \textit{R. Bhatia} [Linear Multilinear Algebra 22, No. 2, 133-147 (1987; Zbl 0641.15007)] conjectured that \(\|\text{diag}(\alpha_1+i\beta_1,\dots,\alpha_n+i\beta _n)\|\leqslant \sqrt 2\|T\|\) for any unitarily invariant norm \(\|\cdot\|\). Here the author proves the above conjecture.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 1331739
zbMATH Keywords: eigenvalue
zbMATH Keywords: singular value
zbMATH Keywords: unitarily invariant norm
zbMATH Keywords: Cartesian decomposition
zbMATH Keywords: norm inequalities
zbMATH Keywords: inequalities involving eigenvalues
reviewed by: A. R. Alimov
MaRDI profile type: Publication
cites work: Eigenvalue inequalities associated with the cartesian decomposition
cites work: Q4040931
Wikidata QID: Q126338675
description: scientific article; zbMATH DE number 1331739