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?