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?

This can always be done for $M$ of even size $2n$, i.e. $M$ is a $2n \times 2n$ matrix.
Set 
$P = \begin{bmatrix} 0 & I_n \\ 0 & 0 \end{bmatrix}, \tag{1}$
where $I_n$ is the $n \times n$ identity matrix.  Then
$P^2 = \begin{bmatrix} 0 & I_n \\ 0 & 0 \end{bmatrix}\begin{bmatrix} 0 & I_n \\ 0 & 0 \end{bmatrix} = 0; \tag{2}$
likewise take
$Q = P^T = \begin{bmatrix} 0 & 0 \\ I_n & 0 \end{bmatrix}; \tag{3}$
then
$Q^2 = (P^T)^2 = P^TP^T = (PP)^T = (P^2)^T = 0 \tag{4}$
as well.  Then
$P + Q =  \begin{bmatrix} 0 & I_n \\ I_n & 0 \end{bmatrix}, \tag{5}$
and
$(P + Q)^2 = \begin{bmatrix} 0 & I_n \\ I_n & 0 \end{bmatrix} \begin{bmatrix} 0 & I_n \\ I_n & 0 \end{bmatrix} = \begin{bmatrix} I_n & 0\\ 0 &  I_n \end{bmatrix} = I_{2n}, \tag{7}$
so setting
$A = sP, \tag{8}$
$B = sQ, \tag{9}$
we find
$(A + B)^2 = s^2(P + Q)^2 = s^2 I_{2n} = M. \tag{10}$
Also worth noting:
$(P + Q)^2 = P^2 + PQ + QP + Q^2 = PQ + QP; \tag{11}$
$PQ = \begin{bmatrix} 0 & I_n \\ 0 & 0 \end{bmatrix}\begin{bmatrix} 0 & 0 \\ I_n & 0 \end{bmatrix} = \begin{bmatrix} I_n & 0 \\ 0 & 0 \end{bmatrix}, \tag{12}$
and likewise
$QP = \begin{bmatrix} 0 & 0 \\ 0 & I_n \end{bmatrix}. \tag{12}$
Remark: Not sure if there is a solution for $M$ of odd size.
