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I am in the process of proving the title.

The hint says, for any polynomial $f$, we have $$f(P^{-1}AP) = P^{-1}f(A)P.$$ A is an $n \times n$ matrix over $F$ while $P$ is an invertible matrix such that the above matrix multiplication $P^{-1}AP$ makes sense.

Why is this true?

Thank you.

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  • $\begingroup$ Hint: What is $(P^{-1}AP)^n$? $\endgroup$ – user60589 Nov 30 '15 at 20:27
  • $\begingroup$ @user60589 It is $P^{-1}A^nP$ $\endgroup$ – user247618 Nov 30 '15 at 20:30
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    $\begingroup$ Can you use this with the fact that $f$ is polynomial? $\endgroup$ – user60589 Nov 30 '15 at 20:34
  • $\begingroup$ That would be useful if the OP wanted to prove the hint - but I think the task is to prove that similar matrices have the same minimal polynomial $\endgroup$ – WW1 Nov 30 '15 at 20:38
  • $\begingroup$ @user60589 aha! got it. Thanks. Also thank you WW1 but I think I know how to proceed from here! $\endgroup$ – user247618 Nov 30 '15 at 20:44
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if $f$ is a polynomial then you have: $$f(x)=a_nx^n+...+a_1x+a_0$$ Then you have $$f(P^{-1}AP)=a_n(P^{-1}AP)^n+...+a_1(P^{-1}AP)+a_0I$$ which is $$f(P^{-1}AP)=a_n(P^{-1}APP^{-1}AP...P^{-1}AP)+...+a_1(P^{-1}AP)+a_0P^{-1}IP$$ or $$f(P^{-1}AP)=P^{-1}a_nA^nP+...+P^{-1}a_1AP+a_0P^{-1}IP$$ which finally gives $$f(P^{-1}AP)=P^{-1}(a_nA^n+...+a_1A+a_0I)P=P^{-1}f(A)P$$

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