By https://en.wikipedia.org/wiki/Nilpotent_matrix, a nilpotent matrix is a square matrix $N$ such that $$ N^{k}=0$$ for some positive integer $k$. I think it is difficult to give a unified expression of the nilpotent matrix. But I am interested in the case of $k=2$.
My question is whether we can give a unified expression of the sequare matrix $N$ such that $N^2=0$ ?
Let $K$ be a field and $N\in M_n(K)$.
Then $N^2=0_n$ IFF there is an invertible $P\in M_n(K)$ and $p\leq n/2$ s.t. $N=PDP^{-1}$ where $D=diag(U_1,\cdots,U_p,0_{n-2p})$ with, for every $i$, $U_i=\begin{pmatrix}0&1\\0&0\end{pmatrix}$.