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Question 7.E.3.8: If f(A) exists for a diagonalizable A, explain why A f(A) = ......

If f(A) exists for a diagonalizable A, explain why A f(A) = f(A)A. What can you say when A is not diagonalizable?

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When A is diagonalizable, (7.3.11) insures f(A) = p(A) is a polynomial in A, and Ap(A) = p(A)A.

G_i = \prod\limits^{k}_{\substack{j=1\\j≠i}} (A  −  λ_jI)/ \prod\limits^{k}_{\substack{j=1\\j≠i}} (λ_i  −  λ_j)      for  i = 1,  2,  .  .  .  ,  k.      (7.3.11)

If f(A) is defined by the series (7.3.7) in the nondiagonalizable case,

f(A) = \sum\limits_{n=0}^{∞} c_n(A  −  z_0I)^n.                        (7.3.7)

then, by Exercise 7.3.7, it’s still true that f(A) = p(A) is a polynomial in A, and thus Af(A) = f(A)A holds in the nondiagonalizable case also.

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