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 and v unit vector. In particular, we establish that with high probability, outlier can be distinguished at all times t>1+N−1∕3+ε, any ε>0. The this natural process combines elements non-Hermitian analysis, illustrates some aspects intrinsic instability (even weakly) matrices.