Joint numerical range and its generating hypersurface
type: Item
instance of: scholarly article
title: Joint numerical range and its generating hypersurface
zbMATH Open document ID: 1185.15021
author: Mao-Ting Chien
author: Hiroshi Nakazato
published in: Linear Algebra and its Applications
publication date: 2009-12-18T00:00:00Z
review text: The authors consider the joint numerical range \(W(H_1, \dots, H_n)\) of \(m\)-tuples of Hermitian matrices. Associated with an~\(m\)-tuple \(H_1, \dots, H_n\) there is the homogeneous polynomial \[ F(y_0,\dots, y_m) = \det (y_0I_n+ y_1H_1 + \dots + y_mH_m), \] which may be looked upon as a~hyperbolic form with respect to the point \((1, 0, 0, \dots, 0) \in{\mathbb R}^{m+1}\). Consider the algebraic variety \[ S_F=\{[(y_0,y_1,\dots, y_m)]\in {\mathbb C\mathbb P}^m: F(y_0,y_1,\dots, y_m)= 0\}. \] The main purpose of this paper is to examine the joint numerical in terms of the hypersurface \(S_F\) and the boundary generating hypersurface \(S_F^\land\). An~example is presented to show that an~analog of Kippenhahn's theorem for the joint numerical range of three Hermitian matrices does not hold.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 32S25
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 5650029
zbMATH Keywords: joint numerical range
zbMATH Keywords: generating hypersurface
zbMATH Keywords: singular point
zbMATH Keywords: Hermitian matrix
zbMATH Keywords: Kippenhahn's theorem
reviewed by: A. R. Alimov
MaRDI profile type: Publication
full work available at URL: j.laa.2009.07.034
OpenAlex ID: W2021887341
cites work: Lacunas for hyperbolic differential operators with constant coefficients.I
cites work: An Extension of the Hausdorff-Toeplitz Theorem on the Numerical Range
cites work: Hermitian Forms and the Fibration of Spheres
cites work: On the Classification of Cubic Surfaces
cites work: Davis-Wielandt shell and \(q\)-numerical range
cites work: Convexity of the joint numerical range: Topological and differential geometric viewpoints.
cites work: Q4040931
cites work: Q3955947
cites work: Über den Wertevorrat einer Matrix
cites work: Sets of matrices with given joint numerical range
cites work: Convexity of the Joint Numerical Range
cites work: On the convex hull of the multiform numerical range
cites work: Q5571936
cites work: Q5799167
DOI: 10.1016/J.LAA.2009.07.034
description: scientific article; zbMATH DE number 5650029