Given matrix $A \in \mathbb{R}^{m \times n}$, are there any results related to its difference array $$A^* \triangleq \left[sign(a_{i,j} + a_{r, s} - a_{r, j} - a_{i, s})\right]_{i<r, j<s}?$$
Or equivalently, given matrix $A \in \mathbb{R}^{m \times n}$, are there any results related to $$\hat{A} \triangleq \left[sign\left(\sum_{x=i}^r \sum_{y=j}^s a_{x, y} \right)\right]_{i\le r, j\le s}?$$ Here, $sign: \mathbb{R} \mapsto \{+, -, 0\}$ is the sign function.
