(Spectral Factorization) Suppose a 3 x 3 matrix A admits a factorization as A = PDP-1 ,...

(Spectral Factorization) Suppose a 3 x 3 matrix A admits a factorization as A = PDP^-1 , where P is some invertible 3 x 3 matrix and D is the diagonal matrix

Show that this factorization is useful when computing high powers of A. Find fairly simple formulas for A², A³, and A^k (k a positive integer), using P and the entries in D.