Abstract

Laura Smithies and Richard S. Varga, Singular value decomposition Geršgorin-type sets, Linear Algebra Appl. 417(2006), 370-380. MR2250318, Zbl. pre 05053757.

Abstract. In this note, we introduce the {\it singular value decomposition Ger\v{s}gorin set}, $\Gamma^{SV}(A)$, of an $N \times N$ complex matrix $A$, where $N \le \infty$. For $N$ finite, the set $\Gamma^{SV}(A)$ is similar to the standard Ger\v{s}gorin set, $\Gamma(A)$, in that it is a union of $N$ closed disks in the complex plane and it contains the spectrum, $\sigma(A)$, of $A$. However, $\Gamma^{SV}(A)$ is constructed using {\em column} sums of singular value decomposition matrix coefficients, whereas $\Gamma(A)$ is constructed using {\em row} sums of the matrix values of $A$. In the case $N = \infty$, the set $\Gamma^{SV}(A)$ is defined in terms of the entries of the singular value decomposition of a compact operator $A$ on a separable Hilbert space. Examples are given and applications are indicated.