  This paper considers $N\times N$ matrices of the form $A_\gamma =A+ \gamma
B$, where $A$ is self-adjoint, $\gamma \in C$ and $B$ is a non-self-adjoint
perturbation of $A$. We obtain some monodromy-type results relating the
spectral behaviour of such matrices in the two asymptotic regimes $|\gamma
|\to\infty$ and $|\gamma |\to 0$ under certain assumptions on $B$. We also
explain some properties of the spectrum of $A_\gamma$ for intermediate sized
$\gamma$ by considering the limit $N\to\infty$, concentrating on properties
that have no self-adjoint analogue. A substantial number of the results extend
to operators on infinite-dimensional Hilbert spaces.
