Singular Value Decomposition Geršgorin Sets

Abstract: In this note, we introduce the singular value decomposition Geršgorin set, Γ (A), of an N ×N complex matrix A, where N ≤ ∞. For N finite, the set Γ (A) is similar to the standard Geršgorin set, Γ(A), in that it is a union of N closed disks in the complex plane and it contains the spectrum, σ(A), of A. However, Γ (A) is constructed using column sums of singular value decomposition matrix coefficients, whereas Γ(A) is constructed using row sums of the matrix values of A. In the case N = ∞, the set Γ (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.