# minimal polynomial is independent of choice of basis

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?

$$M=P^{-1}NP$$ 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 $$N$$.