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Singular value decomposition normally estimated Geršgorin sets. (English) Zbl 1171.15304
Summary: Let $$B\in\mathbb C^{N\times N}$$ denote a finite-dimensional square complex matrix, and let $$V\Sigma W^*$$ denote a fixed singular value decomposition (SVD) of $$B$$. In this note, we follow up work from L. Smithies and R. S. Varga [Linear Algebra Appl. 417, No. 2–3, 370–380 (2006; Zbl 1101.15017)], by defining the SV-normal estimator $$\epsilon_{V\Sigma W^*}$$, (which satisfies $$0\leq\epsilon_{V\Sigma W^*}\leq1$$), and showing how it defines an upper bound on the norm $$\|B^*B-BB^*\|_2$$ of the commutant of $$B$$ and its adjoint $$B^*=\Bar B^T$$. We also introduce the SV-normally estimated Geršgorin set $$\Gamma^{NSV}(V\Sigma W^*)$$ of $$B$$, defined by this SVD. Like the Geršgorin set for $$B$$, the set $$\Gamma^{NSV}(V\Sigma W^*)$$ is a union of $$N$$ closed discs which contains the eigenvalues of . When $$\epsilon_{V\Sigma W^*}$$ is zero, $$\Gamma^{NSV}(V\Sigma W^*)$$ is exactly the set of eigenvalues of $$B$$; when $$\epsilon_{V\Sigma W^*}$$ is small, the set $$\Gamma^{NSV}(V\Sigma W^*)$$ provides a good estimate of the spectrum of $$B$$. We end this note by expanding on an example from Smithies and Varga [loc. cit.], and giving some examples, which were generated using Matlab, of the sets $$\Gamma^{NSV}(V\Sigma W^*)$$ and $$\Gamma^{RNSV}(V\Sigma W^*)$$, the reduced SV-normally estimated Geršgorin set.

MSC:
 15A18 Eigenvalues, singular values, and eigenvectors 47A07 Forms (bilinear, sesquilinear, multilinear)
Zbl 1101.15017
Matlab
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