Exercise 14, § 5

Let $V$ be a finite-dimensional vector space over $K$ and $z$ an endomorphism of $V$. We adopt the notation $z_q^p$ of Exercise 3 of § 3. An endomorphism $z'$ of $V$ is called a replica of $z$ if for all $p$ and $q$ every zero of $z_q^p$ is a zero of $z_q^{p'}$. Show that if $\operatorname{Tr}(zz') = 0$ for every replica $z'$ of $z$, then $z$ is nilpotent. (Use the proof of Lemma 3. In the notation of this proof, prove in particular that $t$ is a replica of $z$.)