Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem

TL;DR

This paper extends the single ring theorem to relate eigenvector correlations and eigenvalue condition numbers in large non-Hermitian matrices, providing a simple formula and numerical validation.

Contribution

It introduces a new formula linking eigenvector correlations with the spectral distribution, extending the single ring theorem to include eigenvalue condition numbers.

Findings

Derived a formula for eigenvector correlations involving spectral distribution

Calculated the conditional expectation of eigenvalue condition numbers

Validated predictions with large-scale numerical experiments

Abstract

We extend the so-called "single ring theorem"[1], also known as the Haagerup-Larsen theorem[2], by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non-hermitian matrix $X$, being the spectral density weighted by the squared eigenvalue condition number, is given by a simple formula involving only the radial spectral cumulative distribution function of $X$. We show that this object allows to calculate the conditional expectation of the squared eigenvalue condition number. We give examples and we provide cross-check of the analytic prediction by the large scale numerics.