On the condition number of the shifted real Ginibre ensemble

TL;DR

This paper provides a precise estimate on the smallest singular value of shifted real Ginibre matrices, revealing how complex shifts improve the condition number decay rate and refining understanding of eigenvalue stability.

Contribution

It offers the first sharp lower tail estimate for the smallest singular value of shifted real Ginibre matrices, confirming a conjecture and improving bounds on eigenvalue condition numbers.

Findings

Shift by complex parameter accelerates decay of condition number tail

Sharp lower tail estimate for smallest singular value derived

Improved bounds on eigenvector overlaps for real Ginibre matrices

Abstract

We derive an accurate lower tail estimate on the lowest singular value $\sigma_1(X-z)$ of a real Gaussian (Ginibre) random matrix $X$ shifted by a complex parameter $z$. Such shift effectively changes the upper tail behaviour of the condition number $\kappa(X-z)$ from the slower $\mathbf{P}(\kappa(X-z)\ge t)\lesssim 1/t$ decay typical for real Ginibre matrices to the faster $1/t^2$ decay seen for complex Ginibre matrices as long as $z$ is away from the real axis. This sharpens and resolves a recent conjecture in [arXiv:2005.08930] on the regularizing effect of the real Ginibre ensemble with a genuinely complex shift. As a consequence we obtain an improved upper bound on the eigenvalue condition numbers (known also as the eigenvector overlaps) for real Ginibre matrices. The main technical tool is a rigorous supersymmetric analysis from our earlier work [arXiv:1908.01653].