I'm trying to work my way through a question I've been set, which is as follows:
(i) For $i$ > 0, let $K_i = KerT^i$. Show that for each i, $K_i$ ⊆$ K_{i+1}$, and hence show that there exists a non-negative integer r such that $K_r = K_{r+1}$. Prove that $K_r= K_{r+j}$ for all j ≥ 1. Hence, or otherwise, show that $V = K_r ⊕ T^r(V)$.
(ii) Suppose that the only eigenvalues of T are $0$ and $λ$, where λ$\neq$0. Let W = $T^r(V)$, with $r$ as above. Show that T(W) $⊆$ W, and that the restriction of T to W has λ as its only eigenvalue. Let S denote the restriction of $(T − λI)$ to W. Show that $0$ is the only eigenvalue of S. By applying (i) with S, W in place of T, V , show that $S^m = 0$ for some $m$.
I've managed to get everything except the very last part, where I'm asked to show that $S^m = 0$ for some $m$. I have no idea what to do so I'd really appreciate it if you could help.
EDIT: I see now that since $0$ is the only eigenvalue of $S$ then the characteristic polynomial's only root will be zero and so the minimal polynomial will end up just being $x^m$ for some $m$, giving the answer, but I don't see how this involves (i). Am I missing something?
$0$ is the only eigenvalue of $S $, which form has the char. polynomial of $S $ ?
Now do not forget Cayley -Hamilton.