Dynamics of a rank-one perturbation of a Hermitian matrix

TL;DR

This paper investigates the evolution of eigenvalues in a time-dependent Hermitian matrix perturbed by a rank-one non-Hermitian component, revealing outlier behavior and instability characteristics.

Contribution

It provides a probabilistic analysis of eigenvalue trajectories in a Hermitian matrix with a rank-one perturbation over time, highlighting outlier detection and matrix instability.

Findings

Outliers can be identified for all times t > 1 + N^{-1/3+ε}

Eigenvalue trajectories exhibit intrinsic instability due to non-Hermitian perturbation

The study bridges Hermitian and non-Hermitian matrix analysis techniques

Abstract

We study the eigenvalue trajectories of a time dependent matrix $ G_t = H+i t vv^*$ for $t \geq 0$, where $H$ is an $N \times N$ Hermitian random matrix and $v$ is a unit vector. In particular, we establish that with high probability, an outlier can be distinguished at all times $t>1+N^{-1/3+\epsilon}$, for any $\epsilon>0$. The study of this natural process combines elements of Hermitian and non-Hermitian analysis, and illustrates some aspects of the intrinsic instability of (even weakly) non-Hermitian matrices.