If A has n independent eigenvectors, we can write A = SΛS−1, where Λ is a diagonal matrix containing the eigenvalues of A. This allows us to easily compute powers of A which in turn allows us to solve difference equations uk+1 = Auk.
Lecture 22: Diagonalization and Powers of A
Problem Solving: Powers of a Matrix
Work the problems on your own and check your answers when you're done.