= Stiemke theorem
{c}
For a real matrix $P$, exactly one of the following holds: there is $\phi$ with $P\phi\geq0$ and $P\phi\ne0$, or there is $q>0$ with $P^Tq=0$. Separating $\operatorname{Im}P$ from the standard simplex proves the result; $q$ may then be normalized to sum to one.
This is an alternative related to <Farkas' lemma>. It is not the strict-primal Gordan alternative: for the column matrix $(1,0)^T$, a nonnegative nonzero primal vector exists, while a componentwise strictly positive one does not.
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