Characterising semi-definite positiveness on vectors with non-negative entries My problem is to characterise (or find useful information on) the cone $C$ of $N\times N$ matrices $M$ ($N\geq 1$) such that $$V^t M V\geq 0$$ for every vector $V $ with non-negative entries. Is this cone of matrices familiar to anyone?
Remark 1: $C$ clearly contains the convex cone $S$ of semi-definite positive matrices, and that of matrices $P$ with non-negative entries. Even though it looks too easy, I could not prove that $C$ is not the convex hull of $S$ and $P$.
Remark 2: My original problem is to characterise the dual cone of $C$, containing the covariances of random vectors of $\mathbb{R}^N$ with a.s. non-negative entries.
 A: (After Noah Stein's answer) By definition, the dual cone $C^\star$ is spanned by matrices $v\otimes v$ with $v\ge0$. The following counter-example is due to Hall. The $5\times5$ symmetric matrix
$$S=\begin{pmatrix}  4 & 0 & 0 & 2 & 2 \\\\ 0 & 4 & 3 & 0 & 2 \\\\ 0 & 3 & 4 & 2 & 0 \\\\ 2 & 0 & 2 & 4 & 0 \\\\ 2 & 2 & 0 & 0 & 4 \end{pmatrix}$$
has non-negative entries and is positive semi-definite. Therefore, it belongs to $(S\cup P)^\star$. Yet, it cannot be written as the sum of $v_j\otimes v_j$ where the vectors $v_j$ are non-negative. Therefore $A\not\in C^\star$. By duality, this proves that $C$ is not the convex hull of $S\cup P$.
The example is analyzed in details in Exercise 347 of my list.
A: Your cone $C$ is the cone of copositive matrices.  The dual of C is the cone of compeltely positive matrices.  See e.g.
http://mathworld.wolfram.com/CopositiveMatrix.html
A: The cone $C$ is called the cone of copositive matrices and its dual $C^*$ is called the cone of completely positive matrices.  Here are some references.
The paper most relevant to your question is probably "On Non-Negative Forms In Real Variables Some Or All Of Which Are Non-Negative," in which P. H. Diananda shows that $C$ does in fact take the simple form you stated in Remark 1 in the case $n\leq 4$.  As mentioned in a note at the end of that paper, a counterexample called the Horn Form shows that $C$ does not have this form when $n\geq 5$.
Another important paper in this area is "Some NP-Complete Problems In Quadratic And Nonlinear Programming" by K. G. Murty and S. N. Kabadi.  There the authors show that checking membership in $C$ is co-NP-complete.  In "Semidefinite programming based tests for matrix copositivity," P. A. Parrilo constructs a hierarchy of outer approximations to $C$ defined via SDPs; their duals give inner approximations to $C^*$.  By combining these ideas one can easily construct semidefinite program approximations to a variety NP-complete optimization problems.
