Spectral properties of random non-self-adjoint matrices and operators

I describe some numerical experiments which determine the degree of spectral instability of medium‐sized 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. I 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. My results imply that the spectrum of the non‐self‐adjoint Anderson model changes suddenly on passing to the infinite volume limit.