On Eigenvector Bounds

There are methods to compute error bounds for a multiple eigenvalue together with an inclusion of a basis of the corresponding invariant subspace. Those methods have no restriction with respect to the structure of Jordan blocks, but they do not provide an inclusion of a single eigenvector. In this note we first show under general assumptions that a narrow inclusion of a single eigenvector is not likely in case of corresponding eigenvalue with geometric multiplicity greater than one. We show a similar result if attention is restricted to symmetric or Hermitian matrices. Finally, we present an inclusion theorem for the eigenvector to an eigenvalue of geometric multiplicity one. Some numerical examples demonstrate the numerical applicability.