Eigenvalue placement for regular matrix pencils with rank one perturbations

TL;DR

This paper analyzes how eigenvalues of regular matrix pencils are affected by rank one perturbations, providing bounds and conditions for eigenvalue placement, with applications to pole placement in differential algebraic equations.

Contribution

It characterizes eigenvalue changes under rank one perturbations and establishes bounds, extending eigenvalue placement theory to matrix pencils with practical applications.

Findings

Eigenvalues of perturbed pencils can be placed arbitrarily within certain bounds.

The maximum number of Jordan chains that can be destroyed is equal to the number of eigenvalues that can be arbitrarily assigned.

Sharp bounds are provided for changes in algebraic and geometric multiplicities under perturbations.

Abstract

A regular matrix pencil sE-A and its rank one perturbations are considered. We determine the sets in the extended complex plane 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 the extended complex plane. 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.