Stiemke theorem

For a real matrix $P$, exactly one of the following holds: there is $\varphi$ with $P\varphi \ge 0$ and $P\varphi \ne 0$, or there is $q>0$ with $P^{T}q=0$. Separating $\mathrm{Im}P$ from the standard simplex proves the result; $q$ may then be normalized to sum to one.