A theory of rank-k (with  perturbation of symplectic matrices and Hamiltonian systems with periodic coefficients using a base of isotropic subspaces is presented. After showing that the fundamental matrix  of the rank-k perturbation of Hamiltonian system with periodic coefficients and the rank-k perturbation of the fundamental matrix  of the unperturbed system are the same, the Jordan canonical form of  is given. Two numerical examples illustrating this theory and the consequences of rank-k perturbations on the strong stability of Hamiltonian systems are also given.

Hamiltonian system, symplectic matrix, isotropic subspace, perturbations, strong stability.