Volume 26, pp. 320-329, 2007.

Singular value decomposition normally estimated Geršgorin sets

Natacha Fontes, Janice Kover, Laura Smithies, and Richard S. Varga

Abstract

Let $B\in {\mathbf{C}}^{N\times N}$ denote a finite-dimensional square complex matrix, and let $V\mathrm{\Sigma }{W}^{\ast }$ denote a fixed singular value decomposition (SVD) of $B$. In this note, we follow up work from Smithies and Varga [Linear Algebra Appl., 417 (2006), pp. 370–380], by defining the SV-normal estimator ${ϵ}_{V\mathrm{\Sigma }{W}^{\ast }}$, (which satisfies $0\le {ϵ}_{V\mathrm{\Sigma }{W}^{\ast }}\le 1$), and showing how it defines an upper bound on the norm, $\Vert B^{\ast }B-B{B}^{\ast }{\Vert }_2$, of the commutant of $B$ and its adjoint, $B^{\ast }={\overline{B}}^T$. We also introduce the SV-normally estimated Geršgorin set, ${\mathrm{\Gamma }}^{NSV}(V\mathrm{\Sigma }{W}^{\ast })$, of $B$, defined by this SVD. Like the Geršgorin set for $B$, the set ${\mathrm{\Gamma }}^{NSV}(V\mathrm{\Sigma }{W}^{\ast })$ is a union of $N$ closed discs which contains the eigenvalues of $B$. When ${ϵ}_{V\mathrm{\Sigma }{W}^{\ast }}$ is zero, ${\mathrm{\Gamma }}^{NSV}(V\mathrm{\Sigma }{W}^{\ast })$ is exactly the set of eigenvalues of $B$; when ${ϵ}_{V\mathrm{\Sigma }{W}^{\ast }}$ is small, the set ${\mathrm{\Gamma }}^{NSV}(V\mathrm{\Sigma }{W}^{\ast })$ provides a good estimate of the spectrum of $B$. We end this note by expanding on an example from Smithies and Varga [Linear Algebra Appl., 417 (2006), pp. 370–380], and giving some examples which were generated using Matlab of the sets ${\mathrm{\Gamma }}^{NSV}(V\mathrm{\Sigma }{W}^{\ast })$ and ${\mathrm{\Gamma }}^{RNSV}(V\mathrm{\Sigma }{W}^{\ast })$, the reduced SV-normally estimated Geršgorin set.

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Key words

Geršgorin type sets, normal matrices, eigenvalue estimates

AMS subject classifications

15A18, 47A07

ETNA articles which cite this article

Vol. 36 (2009-2010), pp. 99-112 Laura Smithies: The structured distance to nearly normal matrices