Exercise 8, § 5

Let $E$ be a vector space of finite dimension $n$ over a field $K$, let $u$ be an endomorphism of $E$, let $V$ be a subspace of $E$ invariant under $u$, and let $u|V$ be the restriction of $u$ to $V$. If $f_i$ ($1 \leq i \leq n$) (resp. $g_j$, $1 \leq j \leq m$) are the similarity invariants of $u$ (resp. $u|V$), show that $g_j$ divides $f_{j + n - m}$ for $1 \leq j \leq m$ (cf. VII, p. 64, Ex. 8, d)). State and prove the analogous property of the endomorphism of $E/V$ induced by $u$.