Recently I have read a proof of the Perron Frobenius Theorem for positive aperiodic matrices. In this proof, the trick was to put a metric in the "positive quadrant" of $\mathbb{R}^n$, $\mathbb{R}^{n}_+$, in order to make the map $\mathbb{R}^n_+\ni p\mapsto pA $ a $\lambda$-contraction, $0<\lambda<1,$ and in this way applies the Contraction Fixed Point Theorem to obtain the only one fixed point for $A.$

To obtain this kind of result, $\mathbb{R}^n_+$ is "projectivezed" and 
the metric to be considerate is the $\textbf{Projective Hilbert Metric}$  [https://en.wikipedia.org/wiki/Hilbert_metric][1]

This proof was given by G. Birkhoff,  [https://en.wikipedia.org/wiki/Garrett_Birkhoff
][1]


  [1]: https://en.wikipedia.org/wiki/Garrett_Birkhoff



**My Question:** What is  the intuition behind the Hilbert Projective metric, there exists a easy way to see that the projective metric $d$ is the right metric  to make $(\mathbb{R}^n_+\ni p\mapsto pA, d) $ a contraction ?