Remark

Let E and F be two finite-dimensional vector spaces over a commutative field K, and let $K'$ be an extension of K. Let $u$ be an endomorphism of E and $v$ an endomorphism of F, and let $u_{(K')}$ and $v_{(K')}$ be the endomorphisms of $E_{(K')}$ and $F_{(K')}$ obtained by extension of scalars. It follows from Corollaries 1 and 2 of VII, §5, No. 3, p. 32 that the endomorphisms $u$ and $v$ are similar if and only if the endomorphisms $u_{(K')}$ and $v_{(K')}$ are. This also follows from Theorem 3 above applied to the algebra A = K[X] and the A-modules $M = E_u$ and $N = F_v$ (VII, §5, No. 1, p. 29).