Numerical ranges of non-normal random matrices: elliptic Ginibre and non-Hermitian Wishart ensembles

TL;DR

This paper characterizes the geometric shape of the numerical range for various non-Hermitian random matrix ensembles, revealing elliptical and non-elliptic structures in the large-system limit.

Contribution

It explicitly describes the limiting numerical range geometries for elliptic Ginibre, chiral, and Wishart ensembles, including products of matrices, advancing understanding of non-normal matrix effects.

Findings

Elliptic Ginibre ensemble's numerical range is an ellipse.

Non-Hermitian Wishart ensemble's numerical range has a non-elliptic envelope.

Product of elliptic Ginibre matrices' numerical range generalizes previous results.

Abstract

The numerical range of a non-normal matrix plays a central role as a descriptor of non-normal effects beyond spectral information. We study a class of fundamental non-Hermitian random matrix ensembles that interpolate between the Hermitian and non-Hermitian regimes. Our analysis focuses on the elliptic Ginibre ensemble and its chiral counterpart, as well as on non-Hermitian Wishart matrices. For each of these models, we explicitly characterise the geometry of the numerical range in the large-system limit. In particular, we show that for the elliptic Ginibre ensemble and its chiral version, the limiting numerical range is an ellipse, whereas for the non-Hermitian Wishart ensemble it is described by a non-elliptic envelope. Furthermore, we determine the numerical range of products of $n$ independent elliptic Ginibre matrices, which recovers, in the cases $n=1$ and $n=2$, the results for the elliptic Ginibre ensemble and the non-Hermitian Wishart ensemble at maximal non-Hermiticity, respectively.