Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 4 / 2F (Topics in Analysis)

The two-dimensional Brouwer fixed-point theorem states that every continuous map from a closed filled triangle to itself has a fixed point. Use the triangle $\Delta =\{x\in {\mathbb{R}}^3:x_i\ge 0,{\sum }_i{x}_i=1\}$, which lies in a two-dimensional affine plane. Strict positivity of the matrix entries ensures $(Ax{)}_i>0$ for every $x\in \Delta$. Thereforeis a continuous self-map of $\Delta$, in fact into its interior. Its fixed point $x$ satisfies $Ax=\lambda x$, where $\lambda ={\sum }_i(Ax{)}_i>0$. Since $x=T(x)$ lies in the interior, $A$ has an eigenvector all of whose entries are strictly positive. This is positive matrix eigenvector from simplex normalization; it uses strict positivity, not merely nonnegative entries.