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Existence of nonnegative solutions to an undetermined underdetermined system of linear equations

Similar questions have been asked elsewhere, but I think this is sufficiently different to warrant a new post. I have a particular matrix $A$ and would like to know when the system $Ax = 0$ has at least one non-negative solution (other than $\vec{0}$). The problem is undeterminedunderdetermined: in most cases I expect the number of variables to be of the order $m^2$, where $m$ is the number of equations. Furthermore, each column of the matrix sums to $0$ and every equation has a mix of positive and negative coefficients. Is this a sufficient condition for the existence of a non-negative solution?

I have seen the algorithm of http://www.jstor.org/pss/1968384, which can be used to test whether a particular system of equations has a non-negative solution, but have not been able to use it to derive a proof for a general family of matrices.

Thanks.
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Existence of nonnegative solutions to an undetermined system of linear equations

Similar questions have been asked elsewhere, but I think this is sufficiently different to warrant a new post. I have a particular matrix $A$ and would like to know when the system $Ax = 0$ has at least one non-negative solution (other than $\vec{0}$). The problem is undetermined: in most cases I expect the number of variables to be of the order $m^2$, where $m$ is the number of equations. Furthermore, each column of the matrix sums to $0$ and every equation has a mix of positive and negative coefficients. Is this a sufficient condition for the existence of a non-negative solution?

I have seen the algorithm of http://www.jstor.org/pss/1968384, which can be used to test whether a particular system of equations has a non-negative solution, but have not been able to use it to derive a proof for a general family of matrices.

Thanks.