Ill-Conditioned Eigensystems and the Computation of the Jordan Canonical Form

The solution of the complete eigenvalue problem for a nonnormal matrix A presents severe practical difficulties when A is defective or close to a defective matrix. Moreover, in the presence of rounding errors, one cannot even determine whether or not a matrix is defective. Several of the more stable methods for computing the Jordan canonical form are discussed, together with the alternative approach of computing well-defined bases (usually orthogonal) of the relevant invariant subspaces.