Eigenstructure of rank one updated matrices

The relationship among eigenvalues of a given square matrix A   and the rank one updated matrix A+vkq⁎A+vkq⁎, where vkvk is an eigenvector of A   associated with the eigenvalue λkλk and q is an arbitrary vector, was described by Brauer in 1952. In this work we study the relations between the Jordan structures of A   and A+vkq⁎A+vkq⁎. More precisely, we analyze the generalized eigenvectors of the updated matrix in terms of the generalized eigenvectors of A, as well as the Jordan chains of the updated matrix. Further, we obtain similar results when we use a generalized eigenvector of A   instead of the eigenvector vkvk.