A Schur decomposition for Hamiltonian matrices

Abstract

A Schur-type decomposition for Hamiltonian matrices is given that relies on unitary symplectic similarity transformations. These transformations preserve the Hamiltonian structure and are numerically stable, making them ideal for analysis and computation. Using this decomposition and a special singular-value decomposition for unitary symplectic matrices, a canonical reduction of the algebraic Riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution. After presenting some real decompositions for real Hamiltonian matrices, we look into the possibility of an orthogonal symplectic version of the QR algorithm suitable for Hamiltonian matrices. A finite-step initial reduction to a Hessenberg-type canonical form is presented. However, no extension of the Francis implicit-shift technique was found, and reasons for the difficulty are given.