A Geometric Perspective on Condensed Forms for Hamiltonian Matrices

Abstract

The solutions of algebraic Riccati equations that arise in linear control theory determine invariant subspaces of Hamiltonian matrices and symplectic matrices. This fact has motivated the pursuit of efficient algorithms for finding eigenvaLues and invariant subspaces of such matrices as a means of efficiently and reliably computing the soLutions of these Riccati equations; see [12], [14], [5], [6], and [13]. Although significant progress has been made in the development of QR-type algorithms for these problems, the initial reduction of the Hamiltonian or symplectic matrix to an appropriate condensed form remains an obstacle to the efficient general application of these algorithms. Recent considerations [3] have cast doubt on the existence of a general finite procedure for performing these reductions.