Sums of Nilpotent Matrices Let $A =$ diag$(a_1,a_2,…,a_n)$, where the sum of all $a_i$’s is zero.
Show that A is a sum of nilpotent matrices.
My idea:
$\begin{bmatrix}
2 & 0 \\
0 & -2
\end{bmatrix}$ = $\begin{bmatrix}
1 & 1 \\
-1 & -1
\end{bmatrix}$ + $\begin{bmatrix}
1 & -1 \\
1 & -1
\end{bmatrix}$
Where $\begin{bmatrix}
1 & 1 \\
-1 & -1
\end{bmatrix}^2$ = $\begin{bmatrix}
1 & -1 \\
1 & -1
\end{bmatrix}^2$ = $0_2$
Extending the 2 $\times$ 2 matrix, we get
$\begin{bmatrix}
2 & 0 & 0 & 0 \\
0 & -2 & 0 & 0 \\
0 & 0 & 2 & 0 \\
0 & 0 & 0 & -2
\end{bmatrix}$ = $\begin{bmatrix}
1 & 1 & 0 & 0 \\
-1 & -1 & 0 & 0 \\
0 & 0 & 1 & 1 \\
0 & 0 & -1 & -1
\end{bmatrix}$ +  $\begin{bmatrix}
1 & -1 & 0 & 0 \\
1 & -1 & 0 & 0 \\
0 & 0 & 1 & -1 \\
0 & 0 & 1 & -1
\end{bmatrix}$
Where $\begin{bmatrix}
1 & 1 & 0 & 0 \\
-1 & -1 & 0 & 0 \\
0 & 0 & 1 & 1 \\
0 & 0 & -1 & -1
\end{bmatrix}^2$ =  $\begin{bmatrix}
1 & -1 & 0 & 0 \\
1 & -1 & 0 & 0 \\
0 & 0 & 1 & -1 \\
0 & 0 & 1 & -1
\end{bmatrix}^2$ = $0_4$
Kindly correct me if I’m wrong, and also, please help me how to generalized this. Thank you in advance.

My answer follows @JoséCarlosSantos' post.
It is enough to notice that for all $1 \leq i < j \leq n$, $$E_{i, i} -E_{j, j} = \left(E_{i, j}\right) + \left(- E_{j, i}\right) + \left(E_{i,  i} -E_{i, j} + E_{j, i} - E_{j, j}\right)$$ and $$\left(E_{i,  i} -E_{i, j} + E_{j, i} - E_{j, j}\right)^{2} = 0$$
