Rank one perturbation with a generalized eigenvector

The relationship between the Jordan structures of two matrices sufficiently close has been largely studied in the literature, among which a square matrix $A$ and its rank one updated matrix of the form $A+xb^*$ are of special interest. The eigenvalues of $A+xb^*$, where $x$ is an eigenvector of $A$ and $b$ is an arbitrary vector, were first expressed in terms of eigenvalues of $A$ by Brauer in 1952. Jordan structures of $A$ and $A+xb^*$ have been studied, and similar results were obtained when a generalized eigenvector of $A$ was used instead of an eigenvector. However, in the latter case, restrictions on $b$ were put so that the spectrum of the updated matrix is the same as that of $A$. There does not seem to be results on the eigenvalues and generalized eigenvectors of $A+xb^*$ when $x$ is a generalized eigenvector and $b$ is an arbitrary vector. In this paper we show that the generalized eigenvectors of the updated matrix can be written in terms of those of $A$ when a generalized eigenvector of $A$ and an arbitrary vector $b$ are involved in the perturbation.