Prove that spec$(f(A)) = f$(spec$(A)).$

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Can someone please explain this proof to me? Thanks!

Let $A \in \mathbb{C}^n$ and let $f(x)$ be a polynomial. Prove that spec$(f(A)) = f$(spec$(A))$ (where if $S \subseteq \mathbb{C},$ $f(S) :=$ $\{ f(s) : s \in S\}$).

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The sophisticated machinery that is Google has this one for us. For a more general answer see the first generated answer, for your specific question see the second generated answer.

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