Positive matrix eigenvector from simplex normalization
= Positive matrix eigenvector from simplex normalization
For a strictly positive real matrix $A$, the map $x\mapsto Ax/\sum_i(Ax)_i$ is a continuous self-map of the probability simplex. A <Brouwer fixed-point theorem> fixed point satisfies $Ax=\lambda x$ with $\lambda>0$. The image lies in the interior of the simplex, so every entry of this <eigenvector> is strictly positive. Nonnegative entries and nonzero columns alone do not imply this strict conclusion.