Elliptic numerical ranges of \(3 \times 3\) companion matrices
type: Item
instance of: scholarly article
title: Elliptic numerical ranges of \(3 \times 3\) companion matrices
zbMATH Open document ID: 1144.15017
author: William Calbeck
published in: Linear Algebra and its Applications
publication date: 2008-05-15T00:00:00Z
review text: The companion matrix associated with a polynomial \[ p(z)=(z-z_1)(z-z_2)(z-z_3)=z^3+az^2+bz+c, \] where \(z_1, z_2, z_3\) are given complex numbers, is \[ A(z_1,z_2,z_3)=\left( \begin{matrix} 0 & 1 & 0\\ 0 & 0 & 1\\ -c & -b & -a \end{matrix}\right). \] It is shown that for any two complex numbers \(z_1\) and \(z_2\) there exists at least one complex number \(z_3\) (and at most five) such that the companion matrix \(A(z_1,z_2,z_3)\) has an elliptic numerical range with foci at \(z_1\) and \(z_2\).
reviewed by: Janko Bračič
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 12D05
zbMATH DE Number: 5275759
zbMATH Keywords: companion matrix
zbMATH Keywords: polynomial
zbMATH Keywords: elliptic numerical range
MaRDI profile type: Publication
full work available at URL: j.laa.2007.12.018
OpenAlex ID: W2023078429
cites work: Elliptic numerical ranges of \(4\times 4\) matrices
cites work: Companion matrices: Reducibility, numerical ranges and similarity to contractions
cites work: Q4040931
cites work: The numerical range of \(3 \times 3\) matrices
cites work: On the number of zeros of certain rational harmonic functions
cites work: Über den Wertevorrat einer Matrix
DOI: 10.1016/J.LAA.2007.12.018
description: scientific article; zbMATH DE number 5275759