Given a matrix $M\in\Bbb F_2^{n\times n}$, define its Hadamard rank $h(M)$ to be the minimum number of rank $\leq2$ matrices in $\Bbb F_2^{n\times n}$ with Hadamard product (that is, the entry-wise product $\circ$) equal to $M$. That is, $$h(M)=\min\{k:\exists M_1,\dots,M_k\mid\max_i\mathrm{rk}(M_i)\le 2,\;M_1\circ M_2\circ\dots \circ M_k=M\}.$$

From Arnaud Mortier's argument, $h(M)\leq n$.

Is there a geometric meaning behind the Hadamard rank $h(M)$?

anyobvious relation to the rank, surely at least faces an uphill battle to justify the name 'corank'. $\endgroup$ – LSpice Jan 12 '18 at 16:464more comments