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

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

My Question What 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 ?