When is the Resolvent Like a Rank One Matrix?

TL;DR

This paper investigates when the resolvent of a matrix closely resembles a rank one matrix, relating this to pseudospectra, eigenvalue neighborhoods, and singular vectors, with theoretical and numerical insights.

Contribution

It establishes a connection between the resolvent's rank-one approximation set and the pseudospectrum, providing new results for Jordan blocks, Toeplitz matrices, and eigenvalue neighborhoods.

Findings

The set where the resolvent approximates a rank one matrix relates to the pseudospectrum.

Disks around eigenvalues are contained in this set, offering new spectral insights.

Numerical experiments show the set of singular vector inner products can be nearly as large as the resolvent approximation set.

Abstract

For a square matrix $A$, the resolvent of $A$ at a point $z \in \mathbb{C}$ is defined as $(A-zI )^{-1}$. We consider the set of points $z \in \mathbb{C}$ where the relative difference in 2-norm between the resolvent and the nearest rank one matrix is less than a given number $\epsilon \in (0,1)$. We establish a relationship between this set and the $\epsilon$-pseudospectrum of $A$, and we derive specific results about this set for Jordan blocks and for a class of large Toeplitz matrices. We also derive disks about the eigenvalues of $A$ that are contained in this set, and this leads to some new results on disks about the eigenvalues that are contained in the $\epsilon$-pseudospectrum of $A$. In addition, we consider the set of points $z \in \mathbb{C}$ where the absolute value of the inner product of the left and right singular vectors corresponding to the largest singular value of the resolvent is less than $\epsilon$. We demonstrate numerically that this set can be almost as large as the one where the relative difference between the resolvent and the nearest rank one matrix is less than $\epsilon$ and we give a partial explanation for this. Some possible applications are discussed.