# Linear Algebra – Minimal polynomial is independent of the choice of base

Suppose I have a linear map $$T: V to V$$and two matrix representations, $$M$$. $$N$$let's say that there is an invertible matrix, $$P$$with the same size and $$M = P ^ {- 1} NP$$, Say $$m_M$$ and $$m_N$$ are minimal polynomials for $$M$$ and $$N$$, How would I show it? $$m_N (M) = 0$$?

May I say that? $$m_N (M) = m_N (P ^ {- 1} NP) = m_N (P ^ {- 1}) m_N (N) m_N (P) = 0$$ and the $$m_M (N) = 0$$ follows in a similar way and therefore is the minimal polynomial independent of the choice of the base?