A diagonalizable nilpotent matrix is zero
= A diagonalizable nilpotent matrix is zero
If a <diagonalizable matrix> satisfies $B^k=0$ for a positive integer $k$, write $B=P\operatorname{diag}(\lambda_j)P^{-1}$. Then every $\lambda_j^k=0$, so every eigenvalue is zero and $B=0$. Diagonalizability is essential: the matrix $\begin{pmatrix}0&1\\0&0\end{pmatrix}$ is nonzero but its square is zero. More generally a nonzero nilpotent operator requires a nontrivial <Jordan block>.