Perturbation Analysis of Hamiltonian Schur and Block-Schur Forms

In this paper we present a complete perturbation analysis for the Hamiltonian Schurform of a Hamiltonian matrix under similarity transformations with unitary symplectic matrices.Both linear asymptotic and non-linear perturbation bounds are presented. The same analysis isalso carried out for two less condensed block-Schur forms. It suggests that the block forms areless sensitive to perturbations. The analysis is based on the technique of splitting operators andLyapunov majorants as well as on a representation of the symplectic unitary group which is convenientfor perturbation analysis of condensed forms. As a corollary a perturbation bound for the stableinvariant subspace of Hamiltonian matrices is obtained. Finally, given an ε-perturbation in the initialHamiltonian matrix, the perturbations in the Hamiltonian Schur form and the unitary symplecticbasis are constructed in the form of power series expansions in ε.