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Say I have a linear map $T:V\to V$, and two matrix representations, $M$, $N$, say, such that there exists an invertible matrix, $P$, with the same dimension and $M=P^{-1}NP$. Say $m_M$ and $m_N$ are minimal polynomials for $M$ and $N$. How would I show that $m_N(M)=0$?

Am I allowed to say that $m_N(M)=m_N(P^{-1}NP)= m_N(P^{-1})m_N(N)m_N(P)=0$ and that $m_M(N)=0$ follows similarly and hence the minimal polynomial is independent of choice of basis?

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$M=P^{-1}NP$ implies $M^k=P^{-1}N^kP$ for all $k$ and so $f(M)=P^{-1}f(N)P$ for all polynomials $f$. Therefore, the set of polynomials that kill $M$ coincides with the set of polynomials that kill $N$.

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