Suppose that obeys the differential equation
where is a constant real matrix.
(i) Suppose that has distinct eigenvalues with corresponding eigenvectors . Explain why may be expressed in the form and deduce by substitution that the general solution of is
where are constants.
(ii) What is the general solution of if , but there are still three linearly independent eigenvectors?
(iii) Suppose again that , but now there are only two linearly independent eigenvectors: corresponding to and corresponding to . Suppose that a vector satisfying the equation exists, where denotes the identity matrix. Show that is linearly independent of and , and hence or otherwise find the general solution of .