Low rank perturbation of Jordan structure

Let A be a matrix and lambda(0) be one of its eigenvalues having g elementary Jordan blocks in the Jordan canonical form of A. We show that for most matrices B satisfying rank (B) less than or equal to g, the Jordan blocks of A + B with eigenvalue lambda(0) are just the g - rank (B) smallest Jordan blocks of A with eigenvalue lambda(0). The set of matrices for which this behavior does not happen is explicitly characterized through a scalar determinantal equation involving B and some of the lambda(0)-eigenvectors of A. Thus, except for a set of zero Lebesgue measure, a low rank perturbation A + B of A destroys for each of its eigenvalues exactly the rank (B) largest Jordan blocks of A, while the rest remain unchanged.