Orthogonality and partial pole assignment for the symmetric definite quadratic pencil Original Research Article

The eigenvectors of a symmetric matrix can be chosen to form an orthogonal set with respect to the identity and to the matrix itself. Similarly, the eigenvectors of a symmetric definite linear pencil can be chosen to be orthogonal with respect to the pair. This paper presents the three sets of matrix weights with respect to which the eigenvectors of the symmetric definite quadratic pencil are orthogonal. One of these 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 differential equations. The solution may be of particular interest in the stabilization and control of flexible, 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.