In a recent interesting paper [1] on the stability of linear mechanical systems S.P. Diliberto discusses certain reduction theorems for symmetric and skew-symmetric Hamiltonian matrices with respect to symplectic orthogonal similarity. In what follows it is shown that a unified simple argument will handle both of these problems together and then a s an example it is indicated how the argument can be also used to obtain the reduction theorem for symplectic orthogonal matrices.