On Hamiltonian and symplectic Hessenberg forms

Abstract

Characterizations are given for the Hamiltonian matrices that can be reduced to Hamiltonian Hessenberg form and the symplectic matrices that can be reduced to symplectic Hessenberg form by orthogonal symplectic similarity transformations. The reduction to these special Hessenberg forms is the missing link in the solution of the open problem of constructing a stable structure-preserving QR-like method of complexity O( n 3) for the computation of invariant subspaces of Hamiltonian and symplectic matrices. Our considerations lead us to propose an approach to the computation of Lagrangian invariant subspaces of a Hamiltonian or symplectic matrix.