On the Change in the Spectral Properties of a Matrix under Perturbations of Sufficiently Low Rank

We show that the $r$ largest Jordan blocks disappear and all other blocks remain the same in the part of the Jordan form corresponding to a given eigenvalue $\lambda$ under a generic rank $r$ perturbation. Moreover, a necessary and sufficient condition on the entries of a perturbation under which the spectral properties of $\lambda$ change in this manner is obtained with the use of the resolvent technique for the case in which the geometric multiplicity of $\lambda$ is greater than or equal to $r$. A Jordan basis in the corresponding root space is constructed from the Jordan chains of the original matrix. A complete description of how the spectrum changes in a small neighborhood of the point $z=\lambda$ is given for the case of a small parameter multiplying the perturbation.