How to express a matrix as sum of two square zero matrices I have a real square matrix $M$ that I'd like to express as $M=A+B$ such that
$A^2=0$,$B^2=0$. $M$ has an additional property that $M^2$ is a scalar matrix :
($M^2=s^2I$); and it's dimension is a power of 2 : $dim(M)=2^n,n>0$; Any suggestions?

Take
$$
A=\begin{pmatrix} 1 & -r^{-1}\cr r & -1 \end{pmatrix},\quad 
B=\begin{pmatrix} 1 & -s^{-1}\cr s & -1 \end{pmatrix}
$$
for non-zero $r,s$, and $M=A+B$.
Then $A^2=B^2=0$ and 
$$
M^2=(A+B)^2=\frac{-r^2+2rs-s^2}{rs}I
$$
