Let $GL(n, \mathbb{R})$ be the group of $n \times n$ invertible matrices in real numbers. Let $G:=G(n, \mathbb{R}_{\geq 0})$ be its subgroup $\{M \in G(n,\mathbb{R}) \mid M, M^{-1} \text{ both have positive entries}\}$. Is there any known results on this group? I am particular interest in how big (roughly) this group could be. I know $S_n \subset G(n, \mathbb{R}_{\geq 0})$ (considering the permutation of a basis of a $n$- dimensional vector space), but I wish it could be much bigger than that.