  A general result on the structure and dimension of the root subspaces of a
matrix or a linear operator under finite rank perturbations is proved: The
increase of dimension from the $n$-th power of the kernel of the perturbed
operator to the $(n+1)$-th power differs from the increase of dimension of the
corresponding powers of the kernels of the unperturbed operator by at most the
rank of the perturbation and this bound is sharp.
