Lemma 7

We have $u_A(\mathbf{R}^n_+)\subset \mathbf{R}_+^n$ if and only if $a_{i,j}\geqslant 0$ for all $i$ and $j$, and $u_A(\mathbf{R}^n_+-\{0\})\subset (\mathbf{R}_+^*)^n$ if and only if $a_{i,j}>0$ for all $i$ and $j$.

Let $(e_1, . . . , e_n)$ be the canonical basis of $\mathbf{R}^n$. The vectors $e_i$ belong to $\mathbf{R}_+^n$ and $u_A(e_i)\in \mathbf{R}_+^n$ for every $i$ (resp. $u_A(e_i)\in (\mathbf{R}_+^*)^n$ for every $i$) if and only if $a_{i,j}\geqslant 0$ for all $i$ and $j$ (resp. $a_{i,j}>0$ for all $i$ and $j$). The result follows by linearity.