Conservative perturbations of positive definite Hamiltonian matrices

We consider Hamiltonian matrices obtained by means of symmetric and positive definite matrices and analyse some perturbations that maintain the eigenvalues on the imaginary axis of the complex plane. To obtain this result we prove for such matrices the existence of a diagonal form or, alternatively by means of symplectic transformations, the existence of the simplest canonical form. Applications related to a pair of problems in the context of linear algebra and differential equations are also reported.