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This problem is related to exercise 18(a) in Section 4.4, so you might want to look at that problem for suggestions.
Let A be an n-by-n matrix, and suppose that λ is an eigenvalue with corresponding eigenvector x: so, suppose that A x = λ x. Suppose that f(t) is a polynomial:
f(t) = a0 + a1 t + a2 t2 + ... ak tk.
Show that f(A) x = f(λ) x.
(The n-by-n matrix f(A) is defined just as q(H) is in exercise 18, Section 4.4.)