Biorthogonality and Partial Pole Assignment for the Symmetric Definite Quadratic Pencil

The eigenvectors of a symmetric matrix can be chosen to form a biorthogonal set with respect to the identity and to the matrix itself. Similarly, the eigenvectors of a symmetric de nite linear pencil can be chosen to be biorthogonal with respect to the pair.This paper presents the three sets of matrix weights, with respect to which the eigenvectors of the symmetric de nite quadratic pencil are biorthogonal. One of these relations is used to derive an explicit solution of the partial pole assignment problem by state feedback control for a control system modeled by a system of second order di erential equations. The solution may be of particular interest in the stabilization and control of exible, large, space structures where only a small part of the spectrum is to be reassigned and the rest of the spectrum is required to remain unchanged.