  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.
