Solution
= Solution
On the simplex define $f(x)=Ax/(\mathbf1^TAx)$. Strict positivity makes the denominator positive and maps the simplex continuously into its interior. By <Brouwer fixed-point theorem>, $f(x)=x$ for some $x$. Thus $Ax=λx$, where $λ=\mathbf1^TAx>0$, and because $Ax$ is strictly positive, so is $x$.