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Given $A \in M_{n \times n} (\mathbb C)$ with characteristic polynomial $P_A(x) = (x-\lambda_1)^{n_1} \cdot ... \cdot (x-\lambda_k)^{n_k}$. Let $g \in \mathbb {C [x]}$ a polynomial. Calculate the characteristic polynomial of $g(A)$.

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Hint: suppose first that $A$ is diagonal. –  Daniel Fischer Jul 16 '13 at 15:51

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