Thank you very much, that indeed solves the problem. Unfortunately I was a bit tired yesterday and I didn't state my question correctly. What I really have are $m$ matrices $A_j$ $1 \leq j \leq m$ and I am looking for a single positive matrix $L$ and vectors $u_j$ so that $\sum_j A_j u_j \geq 0$ and $Lu_j = u_j - y_j$, where $y_j$ is given. I'm not sure if I should post a new question or edit the one above.

Thank you, I should have seen that. Let's assume for now that there is no restriction on what the columns sum to.