I have a data matrix $X \in \mathbb{R}^{n \times m}$, with $\mathrm{rank}(X) = n$ and $n \leq m$. I'm trying to understand if I can find some square matrix $C \in \mathbb{R}^{m \times m}$ such that
$$ \left\{ \begin{aligned} & X C = 0,\\ & C^\top \mathbb{1}_m \leq \mathbb{1}_m,\\ & C_i \geq 0,\, i = 1,\ldots,m, \end{aligned} \right. $$
where $C_i$ is the $i$-th column of $C$, while $\mathbb{1}_m$ is a column vector of ones of dimension $m$. Obviously, $C = 0$ does the job. Can anyone see some other possibility, or $C = 0$ is the unique solution?
There exist matrices $X$ for which $C = 0$ is the unique solution and others for which it is not.
Note that $C$ satisfies this condition if and only if each column $v$ of $C$ satisfies the conditions $$ \begin{cases} Xv = 0,\\ 1_m^T v \leq 1,\\ v \geq 0. \end{cases} $$ So, it is equivalent to determine the conditions under which the above equation has a solution.
In fact, a non-zero solution to this solution exists if and only if there exists a non-zero first and third conditions, namely $$ \begin{cases} Xv = 0,\\ v \geq 0. \end{cases} $$ I don't know of any useful alternative formulation for these conditions together with $v \neq 0$.