Definite triples of Hermitian matrices and matrix polynomials
type: Item
instance of: scholarly article
title: Definite triples of Hermitian matrices and matrix polynomials
zbMATH Open document ID: 1047.15009
DOI: 10.1016/S0377-0427(02)00736-7
published in: Journal of Computational and Applied Mathematics
publication date: 2003-03-16T00:00:00Z
review text: Given nonzero \(n\times n\) Hermitian matrices \(A\), \(B\), and \(C\), the joint numerical range of the triple \((A,B,C)\), that is, the set \(F(A,B,C)=\{(x^*Ax,x^*Bx,x^*Cx)\in \mathbb{R}^n: x\in \mathbb{C}^n,\;x^*x=1\}\) is convex for \(n\geq 3\) and, it is either convex or the surface of an ellipsoid for \(n=2\). The main result of this paper is a formula for the joint inner numerical radius of a general triple \((A,B,C)\), that is, the distance between \((0,0,0)\) and the boundary of \(F(A,B,C)\). An algorithm for testing definiteness and a study of the numerical ranges for special matrix polynomials are also included.
reviewed by: Aurelian Gheondea
Mathematics Subject Classification ID: 15A22
Mathematics Subject Classification ID: 15A54
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 1882240
zbMATH Keywords: Crawford number
zbMATH Keywords: gyroscopic system
zbMATH Keywords: joint numerical range
zbMATH Keywords: matrix polynomial
zbMATH Keywords: Hermitian matrices
zbMATH Keywords: algorithm
zbMATH Keywords: joint inner numerical radius
zbMATH Keywords: quasihyperbolic polynomial
author: Panayiotis J. Psarrakos
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1882240