Is there a kind of matrices in $[-1,1]^{n\times n}$ that, when we sum two of them, we obtain a Hadamard matrix of order $n$ (assumed that $n$ is $1$, $2$ or divisible by $4$) as their sum?
For example, we can obtain a (more general) matrix $S\in\{-1,1\}^{m\times n}$ as a sum of two matrices $A,B\in[-1,1]^{m\times n}$ by defining $B=A-1_{m\times n}$.
Edit: As I made some progress on understanding the basic problem I face, I'm able to specify my question much more. I'm specifically trying to express a Hadamard matrix of order $n$ as a sum of two matrices $A,B\in[-1,1]^{n\times n}$ such that at least one of them must have decaying singular values (i.e., it's $k$ last singular values, for $k=1,\dots ,m \, , m\ll n$, according to their ordering, must have a very small magnitude, nearly close to zero).