Eigenvalue Placement for Regular Matrix Pencils with Rank One Perturbations

A regular matrix pencil $sE-A$ and its rank one perturbations are considered. We determine the sets in $\mathbb{C}\cup\{\infty\}$ which are the eigenvalues of the perturbed pencil. We show that the largest Jordan chains at each eigenvalue of $sE-A$ may disappear and the sum of the length of all destroyed Jordan chains is the number of eigenvalues (counted with multiplicities) which can be placed arbitrarily in $\mathbb{C}\cup\{\infty\}$. We prove sharp upper and lower bounds of the change of the algebraic and geometric multiplicity of an eigenvalue under rank one perturbations. Finally we apply our results to a pole placement problem for a single-input differential-algebraic equation with feedback.