Low Rank Perturbation of Jordan Structure

Let A be a matrix and λ0 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) ≤ g, the Jordan blocks of A+B with eigenvalue λ0 are just the g− rank (B) smallest Jordan blocks of A with eigenvalue λ0. 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 λ0-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.