Rank one perturbation with a generalized eigenvector

TL;DR

This paper investigates how generalized eigenvectors of a matrix change under rank-one perturbations involving a generalized eigenvector, extending classical results to more general cases without restrictions on the perturbation vector.

Contribution

It provides a new characterization of generalized eigenvectors of a rank-one perturbed matrix when the perturbation involves a generalized eigenvector, removing previous restrictions on the perturbation vector.

Findings

Generalized eigenvectors of the perturbed matrix can be expressed in terms of those of the original matrix.

The results extend classical eigenvalue perturbation theory to generalized eigenvectors.

The paper removes restrictions on the perturbation vector in the context of generalized eigenvectors.

Abstract

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.