  We consider perturbations of a large Jordan matrix, either random and small
in norm or of small rank. In both cases we show that most of the eigenvalues of
the perturbed matrix are very close to a circle with centre at the origin. In
the case of random perturbations we obtain an estimate of the number of
eigenvalues that are well inside the circle in a certain asymptotic regime. In
the case of finite rank perturbations we completely determine the spectral
asymptotics as the size of the matrix increases. The paper provides an
elementary illustration of some standard techniques of spectral theory.
