Suppose a 3 x 3 matrix A admits a factorization as A = PDP - , where

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.

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1 D = | 0 0 0 1/2 0 0 0 1/3