In studying the existence and
smoothness of invariant manifolds arising from nonlinear, perturbed systems of
ordinary differential equations, one encounters the study of certain linear (in x),
perturbation problems of the type
𝜃
= a + 𝜖b(𝜃,𝜖)
ẋ
= (A + 𝜖B(𝜃,𝜖))x
where 𝜃 and x are vectors, A and B are matrices, b and B are multiply periodic in 𝜃,
and 𝜖 is a perturbation parameter. Assuming A is a constant matrix consisting of
square sub-matrices on the diagonal,
with the maximum of the real parts of the eigenvalues of Ajj less than the minimum
of the real parts of the eigenvalues of Akk for 1 ≦ j < k ≦ n; we construct a change of
variables which reduces B to similar diagonal form.
