A result complementary to Geršgorin's circle theorem
type: Item
instance of: scholarly article
title: A result complementary to Geršgorin's circle theorem
zbMATH Open document ID: 1176.15025
author: Beresford N. Parlett
published in: Linear Algebra and its Applications
publication date: 2009-06-04T00:00:00Z
review text: \textit{S. Geršgorin's} theorem [Izv. Akad. Nauk SSSR, Otd. Mat. Estest. Nauk, VII. Ser. No. 6, 749--754 (1931; Zbl 0003.00102)] about circles in the complex plane containing an eigenvalue of a complex matrix \(M\), though simple to prove, has given rise to a large area known as localization theory, on which \textit{R. S. Varga} has written a book [Geršgorin and his circles. Springer Series in Computational Mathematics 36. Berlin: Springer (2004; Zbl 1057.15023)]. In this paper the author proves a result which is complementary to Geršgorin's theorem in two ways. First, it concerns eigenvectors of \(M\) rather than eigenvalues, and secondly, it examines the consequences of the corresponding eigenvalue not lying in a Geršgorin circle. Though elementary, the result yields an economical way of determining where an eigenvector of a narrow banded matrix is negligible, and finds application in physics research on the ``localization'' of eigenvectors.
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 65F15
zbMATH DE Number: 5561606
zbMATH Keywords: Geršgorin's circle theorem
zbMATH Keywords: eigenvector
zbMATH Keywords: eigenvalue
describes a project that uses: PARDISO
reviewed by: Rabe von Randow
MaRDI profile type: Publication
full work available at URL: j.laa.2009.01.030
OpenAlex ID: W2051386052
cites work: Q4479208
cites work: On Large-Scale Diagonalization Techniques for the Anderson Model of Localization
cites work: Q4829998
DOI: 10.1016/J.LAA.2009.01.030
description: scientific article; zbMATH DE number 5561606