  We describe some numerical experiments which determine the degree of spectral
instability of medium size randomly generated matrices which are far from
self-adjoint. The conclusion is that the eigenvalues are likely to be
intrinsically uncomputable for similar matrices of a larger size. We also
describe a stochastic family of bounded operators in infinite dimensions for
almost all of which the eigenvectors generate a dense linear subspace, but the
eigenvalues do not determine the spectrum. Our results imply that the spectrum
of the non-self-adjoint Anderson model changes suddenly as one passes to the
infinite volume limit.
