Dynamics of a rank-one perturbation of a Hermitian matrix

Abstract

We study the eigenvalue trajectories of a time dependent matrix $G_t=H+itv{v}^{\ast }$ for $t\ge 0$, 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 $t>1+N^{-1∕3+\mathit{\epsilon }}$, for any $\mathit{\epsilon }>0$. 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.