Inclusion regions for matrix eigenvalues
type: Item
instance of: scholarly article
title: Inclusion regions for matrix eigenvalues
zbMATH Open document ID: 1054.15020
DOI: 10.1016/S0024-3795(02)00279-3
author: Ilse C. F. Ipsen
author: Christopher Beattie
published in: Linear Algebra and its Applications
publication date: 2003-02-05T00:00:00Z
review text: The authors derive inclusion regions that contain at least \(j\) eigenvalues of a complex matrix \(A\). When \(A\) is Hermitian, these reduce to the regions discussed by \textit{N. Lehmann} [Zur Verwendung optimaler Eigenwerteingrenzungen bei der Lösung symmetrischer Matrixenaufgaben, Numer. Math. 8, 42--55 (1966; Zbl 0143.37505) and Berechnung von Eigenwertschrauben bei linearen Problemen, Arch. Math. 2, 139--147 (1949)]. Let \(A\) be \(n\times n\), let \(S\) be an \(n\times m\) matrix with orthonormal columns, let \(\tau_1\leq\tau_2\leq\cdots\leq \tau_m\) be the singular values of \((A-\rho I)S\), and let \(\rho\) be a complex number. If \(A\) is diagonalizable and \(A= X\Delta X^{-1}\) is an eigenvalue decomposition, then for each \(j\), \(1\leq j\leq m\), at least \(j\) eigenvalues of \(A\) lie in the disk \(\{z:| z-\rho|\leq \text{cond}(X)\tau_{-j}\}\). If \(A\) is not normal, at least \(j\) eigenvalues of \(A\) lie in the disk \(\{z:| z-\rho|\leq \tau_{-j}+\delta(A)\}\). Here \(\text{cond}(X)= \| X\|\,\| X^{-1}\|\) where \(\| X\|\) denotes the 2-norm, and \(\delta(A)\) denotes the departure from normality of \(A\). Examples show these disks are optimal for normal matrices, but if \(A\) is normal and the range of \(S\) is far from an invariant subspace of \(A\), or if \(A\) is not normal, these inclusion regions can be arbitrarily pessimistic. The authors also derive inclusion regions that are exteriors of disks and annular exclusion regions, when \(S\) has orthonormal columns. When those columns are not orthonormal, they derive inclusion regions involving generalized singular values of matrix pairs.
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 65F15
zbMATH DE Number: 1864801
zbMATH Keywords: eigenvalue inclusion regions
zbMATH Keywords: Hermitian matrix
zbMATH Keywords: normal matrices
zbMATH Keywords: singular values
MaRDI profile type: Publication
cites work: Harmonic Ritz and Lehmann bounds
cites work: Q5689624
cites work: Generalizations of Temple's Inequality
cites work: Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices
cites work: Matrix Analysis
cites work: Zur Verwendung optimaler Eigenwerteingrenzungen bei der Lösung symmetrischer Matrizenaufgaben
cites work: Q3868672
cites work: Q3998482
full work available at URL: s0024-3795(02)00279-3
OpenAlex ID: W1972331617
description: scientific article; zbMATH DE number 1864801