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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)
Citations:
Zbl 1101.15017
Software:
Matlab
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