2. Consider n x n matrices A, D, P, P-1 where P-1 indicates the inverse of P and D is a diagonal matrix, the only nonzero entries are on the diagonal, d¿¿ with di; = 0 if i ‡ j. (a) Assume that A = PDP and hence AP = PD. Explain why each column of P is an eigenvector of A. That is, if x is the jth column of P then Ax = λ for some number (the eigenvalue). What is \ in terms of entries of D? (b) Prove by induction that A = PDP-1.

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