## Singular value decomposition normally estimated Geršgorin sets

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

### Abstract

Let $B \in {\bf C}^{N \times N}$ 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 Smithies and Varga [Linear Algebra Appl., 417 (2006), pp. 370–380], by defining the SV-normal estimator $\epsilon_{V \Sigma W^*}$, (which satisfies $0~\le~\epsilon_{V \Sigma W^*}~\le~1$), and showing how it defines an upper bound on the norm, $\| B^* B - B B^* \|_2$, 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 $B$. 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 [Linear Algebra Appl., 417 (2006), pp. 370–380], 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.

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### Key words

Geršgorin type sets, normal matrices, eigenvalue estimates

15A18, 47A07