Proposition 10

— Let $E$ be a vector space of finite dimension $n$ over a commutative field $K$, let $u$ be an endomorphism of $E$, and let $\chi_u(X) = \prod_{i=1}^n (X - \alpha_i)$ be a decomposition into linear factors of its characteristic polynomial (in a suitable extension of $K$, cf. V, p. 21). If $q$ is a polynomial with coefficients in $K$, then the characteristic polynomial of $q(u)$ is given by

(6)

and its trace and determinant are given by

(7)

(8)

It is clear that (7) and (8) follow from (6) by virtue of (5). To prove the formula (6), we may assume that K is algebraically closed. We then take a basis for E with respect to which the matrix $U$ of $u$ is lower triangular ($VII$, p. 35, Cor. to Prop. 8); we will make use of the following easy lemma: