Consider $U=[u_1 ... u_N]$ to be a non-singular square matrix. I am interested in range space of $U$ but with one caveat, I want to find all $a$ such that each element of $Ua$ is non-negative. In other words for all natural basis vectors $e_i$
$0\leq e_i^TUa, \forall i$
Is there a non-graphical, non-numerical approach that I can take to obtain all possible $a$ (hopefully with a polynomial complexity)?
Edit: $a$ are column vectors, to eliminate scaled solutions of an $a$ vector a normalization condition may be assumed: i.e. $a^Ta=1$
Let $$\Bbb R^N_+=\{x=(x_1,\dots, x_N)\in\Bbb R^N: x_i\ge 0\mbox{ for each }i\}=$$ $$\{\lambda_1e_1+\dots+\lambda_NeN:\lambda_i\ge 0\mbox{ for each }i\}.$$ Since the matrix $U$ is non-singular, $$U^{-1}(\Bbb R^N_+)=\{\lambda_1U^{-1}e_1+\dots+\lambda_NU^{-1}e_N:\lambda_i\ge 0\mbox{ for each }i\}.$$