From this Wikipedia link.
Let $K$ be a field of characteristic zero and Let $A$ be an $n \times n$ matrix over $K$.Prove the following facts:
(a), $N$ is nilpotent iff Trace$(N^m)=0$ for all $0
(b), If $N$ is nilpotent then $N$ is similar to a strict upper triangular matrix.
Additional Query: Is above true if $K$ is is not assumed to be algebraically closed?(Then one can't apply Jordan normal form anymore)