Linear Algebra eigenvalue proof question?

Let A be an nxn matrix, and let v be an eigenvector of A with eigenvalue λ. Prove that if f(t) is any polynomial, then f(A)v=f(λ)v.

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  • Eugene
    Lv 7
    6 years ago
    Favorite Answer

    Write f(t) = ∑(i = 0, k) c_i t^i Since Av = λv,

    A^i(v) = A^(i-1)(Av) = λA^(i-1)v = ••• = λ^(i-1)Av = λ^i(v)

    for all i ≥ 1. Therefore

    f(A)v = ∑(i = 0, k) c_i A^i(v) = ∑(i = 0, k) c_i λ^i(v) = f(λ)v.

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