Let $M$ be a real symmetric positive definite matrix. Can we characterize the sign vectors $x$, that satisfy the condition: $$x_i = \operatorname{sign}\left(\sum_j M_{ij}x_j\right)$$ That is, this says that applying $M$ to the sign vector $x$, returns a vector in the same orthant as $x$, since $$x_i\sum_j M_{ij}x_j \ge 0$$ How many such sign vectors can there be? I have the intuition it's no more than $2n$ (the dimension of the space), but I have no idea how to prove this.
Note: I'm not sure if the positive definiteness of $M$ is helpful here or not, but in the context where I encountered this problem, $M$ is posdef so I added that condition. But it could be irrelevant.