Perturbation Analysis of Hamiltonian Schur and Block-Schur Forms
Μ.Μ. Konstantinov, Volker Mehrmann, Petko H. Petkov · SIAM Journal on Matrix Analysis and Applications · 2001
In this paper we present a complete perturbation analysis for the Hamiltonian Schur form of a Hamiltonian matrix under similarity transformations with unitary symplectic matrices. Both linear asymptotic and nonlinear perturbation bounds are presented. The same analysis is also carried out for two less condensed block-Schur forms. It suggests that the block forms are less sensitive to perturbations. The analysis is based on the technique of splitting operators and Lyapunov majorants as well as on a representation of the symplectic unitary group which is convenient for perturbation analysis of condensed forms. As a corollary, a perturbation bound for the stable invariant subspace of Hamiltonian matrices is obtained. Finally, given an ${\varepsilon}$-perturbation in the initial Hamiltonian matrix, the perturbations in the Hamiltonian Schur form, and the unitary symplectic basis are constructed in the form of a power series expansion in ${\varepsilon}$.