Dynamics of a rank-one perturbation of a Hermitian matrix

We study the eigenvalue trajectories of a time dependent matrix Gt=H+itvv∗ for t≥0, where H is an N×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+ε, for any ε>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.