Chapter 2.5, Problem 26E

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

$D=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1/2& 0\\ 0& 0& 1/3\end{array}\right]$

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.