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Sorry my question was not clearly stated. I will ask it more clearly.

Let $G$ be a matrix with only nonnegative elements with linearly independent columns. Then there exists a column, ${\bf g}$ of $G$ such that the orthogonal projection of ${\bf g}$ on the remaining columns of G is a nonnegative linear combination of those columns. In other words, for a suitable column ${\bf g}$ of $G$ the vector ${\bf x}$ that minimizes $\parallel G^* {\bf x} - {\bf g}\parallel$ has nonnegative elements, where $G^*$ is a matrix consisting of all the columns of $G$ except ${\bf g}$.

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  • $\begingroup$ What do you mean by "projection"? If you mean orthogonal projection, then the unit matrix is a counterexample, but the only one. $\endgroup$
    – user1688
    Commented Dec 26, 2012 at 20:57
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    $\begingroup$ Voting to close as a duplicate of mathoverflow.net/questions/117278/… $\endgroup$
    – Zev Chonoles
    Commented Dec 26, 2012 at 20:58

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What do you mean by "positive linear combination"?

Try G=I. If you select any column of G, you'll find that its projection onto the space spanned by the other columns of G is the 0 vector.

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  • $\begingroup$ Thank you for your response. I tried to clarify the question but I think my question got closed. Since I am new here, I am not sure what to do about that. However, to respond to your question, I meant actually a linear combination with nonnegative coefficients. So basically you have a matrix with positive elements and you want to show the existence of a column, which when orthogonally projected on the remaining columns, can be represented as a nonnegative linear combination of those columns. $\endgroup$
    – Hasan Kurusogan
    Commented Dec 26, 2012 at 21:32
  • $\begingroup$ Sorry about this, I always mean nonnegative when I say positive. $\endgroup$
    – Hasan Kurusogan
    Commented Dec 26, 2012 at 21:33

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