Nilpotence from vanishing traces of powers ($\mathrm{tr}(A^k)=0(1\le k\le n)⟹A^n=0$)

For a complex $n$ by $n$ matrix, triangularization gives $\mathrm{tr}(A^k)={\sum }_j{\lambda }_j^k$. The zero power sums force a finite complex multiset to vanish criterion makes every diagonal eigenvalue zero. A strictly upper-triangular $n$ by $n$ matrix has zero $n$th power: every nonzero product entry would require $n$ strictly increasing index steps. Similarity then gives $A^n=0$.