Typical changes in spectral properties under perturbation by an operator of rank one. (Q1889573)

The author investigates the typical changes in spectral properties under perturbation by operators of rank one. He shows that, for a square matrix, a fixed eigenvalue loses one Jordan block of maximal order under perturbation by a one rank operator, while the orders of the other Jordan blocks remain unchanged. For such a perturbed matrix, new eigenvalues as well as the corresponding eigenvectors are constructed.