Is this a linear optimization problem? $Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative

Question

$Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative.

What should $A$ satisfy to guarantee the equation set have only zero solution?

Answer 1

If a nonzero $x\geq 0$ is a solution to $Ax=0$ then $\sum_j x_j=a>0$, and $x$ is also a solution to the system $$A x=0,\quad 0\leq x,\quad e^\top x\leq a,\qquad\qquad\qquad\qquad(*) $$ where $e$ denotes the all-1 vector. It is a standard application of Farkas lemma in an appropriate form (one might want to use instead $e^\top x=a$) to check whether $(*)$ has a solution. Or, indeed, you may maximise $\sum_j x_j$ subject to $(*)$ and see whether it has maximum $a$ or $0$.

Answer 2

The paper "Conditions for a Unique Non-negative Solution to an Underdetermined System" seems to answer your question.