On symmetric functions of the eigenvalues of the sum of two Hermitian matrices
type: Item
instance of: scholarly article
title: On symmetric functions of the eigenvalues of the sum of two Hermitian matrices
zbMATH Open document ID: 0756.15026
DOI: 10.1016/0024-3795(92)90219-Z
published in: Linear Algebra and its Applications
publication date: 1993-01-16T00:00:00Z
review text: Es seien \(A\) und \(B\) Hermitesche \(n\times n\)-Matrizen mit Eigenwerten \(a_ 1,\dots,a_ n\) bzw. \(b_ 1,\dots,b_ n\), und \(t_ 1,\dots,t_ n\) seien die Eigenwerte der Summenmatrix \(T=A+B\). Verf. untersucht, für welche symmetrischen Funktionen \(f\) der Eigenwerte \(f(t_ 1,\dots,t_ n)\) in der konvexen Hülle von \(\{f(a_ 1+b_{\pi_ 1},\dots,a_ n+b_{\pi n}):\pi\in S_ n\}\) liegt, und beweist, daß dies im Fall \(f(t_ 1,\dots,t_ n)=\sum\{t^ m_ j:j=1,\dots,n\}\) mit beliebigem \(m\in\mathbb{N}\), \(m\geq 2\) und auch im Fall \(f(t_ 1,\dots,t_ n)=\Pi\{\lambda+t_ j:j=1,\dots,n\}\) zutrifft. Im zweiten Fall ist dabei die konvexe Hülle im Raum der reellen Polynome \(n\)-ten Grades zu bilden.
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 95498
zbMATH Keywords: symmetric functions
zbMATH Keywords: eigenvalues
zbMATH Keywords: sum of two Hermitian matrices
zbMATH Keywords: convex hull
author: S. W. Drury
reviewed by: Hans-Joachim Kowalsky
MaRDI profile type: Publication
cites work: On the determinantal conjecture of Marcus and de Oliveira
cites work: Bounds for the Determinant of the Sum of Hermitian Matrices
cites work: Fonctions composees différentiables
cites work: Q3293417
cites work: Q4040931
cites work: Q3954445
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description: scientific article; zbMATH DE number 95498