Let $A$ and $B$ be in $\mathcal{M}_{n}(\{-1,1\})$. Suppose that there exist diagonal matrices $C$ and $D$ containing only 1s and $-1$s on the diagonal, such that $B = CAD$. Show that $S(A) = S(B)$.
Let $A$ and $B$ be in $\mathcal{M}_{n}(\{-1,1\})$. Suppose that there exist diagonal matrices $C$ and $D$ containing only 1s and $-1$s on the diagonal, such that $B = CAD$. Show that $S(A) = S(B)$.