Given $(0,1)$-matrices $A_1,A_2,...,A_k$ of size $m \times n$, I am interested in finding scalars $c_1,c_2,...,c_k\geq1$ such that $Rank(c_1 A_1+c_2 A_2+...+c_k A_k)=1$. I have several questions:
Are there necessary or sufficient conditions we can give for the existence of a solution?
When a solution does exist, how can I find one?
In particular, I would like to find a solution that minimizes $\vec{c}$ with respect to a norm (1-norm would be best, but I'm not too picky at this point). Any suggestions for doing this?
If it helps: $k\leq20$ and $m,n\leq6$.