Let 
$$A = \left(
\begin{array}{ccc}
\sum_{j\ne 1}a_{1j} & -a_{12} & \cdots & -a_{1n}\\
-a_{21} &  \sum_{j\ne 2}a_{2j} & \cdots & -a_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
-a_{n1} & -a_{n2} & \cdots & \sum _{j\ne n}a_{nj}\\
\end{array}\right)
$$ be an $n$ by $n$ symmetric matrix, i.e.  for all distinct $i,j \in \{1,2, \ldots, n\}$, we have that $a_{ij} = a_{ji}$.

Here is the question: Is there any nice necessary and sufficient condition on positive semi-definiteness of $A$.

*)Note that he non-negativity of $a_{ij}$s is a sufficient condition, since then $A$ is a diagonally dominant matrix.
 
**)Also note that if $f({\rm x}) = {\rm x}^T A {\rm x}$, then 
$
f({\rm x}) = \sum_{i<j} a_{ij} (x_i - x_j)^2
$. Considering this quadratic form might be useful.