Corollary 2

— In the notation of Cor. 1, let $q(X)$ be the minimal polynomial of $u$; then $q(X)$ divides $\chi_u(X)$ and $\chi_u(X)$ divides $q(X)^n$. In particular the minimal polynomial and the characteristic polynomial of $u$ have the same roots, and these are the eigenvalues of $u$.

Since $q(X) = q_n(X)$, it is clear that $q(X)$ divides $\chi_u(X)$. On the other hand, since each $q_i$ divides $q$, their product $\chi_u$ divides $q^n$.