Low Rank Perturbation of Jordan Structure
Julio Moro, Froilán M. Dopico · SIAM Journal on Matrix Analysis and Applications · 2003
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 ${\rm rank}\,(B)\leq g$, the Jordan blocks of A+B with eigenvalue $\lambda_0$ are just the $g-{\rm rank}\,(B)$ 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.