I derive this question while trying to prove the monotonicity of a differentiable vector function $f(x)$ that maps from $X\subset R^n$ to $R^n$. The domain $X$ only consists of vectors $x$ such that $1'x=0$, here $1$ is the vector of all ones.

Using the mean-value theorem, we have that $f(x)$ is locally monotone at $x$ if its Jacobian matrix evaluated at $x$, which we label as $A$, satisfies the following condition:

$$v'Av\geq 0,\quad \forall v \text{ such that } 1'v=0.$$

This is a weaker condition than positive semidefiniteness. However, while there are a number of ways to characterize positive semidefinite matrices, for example, see this Wikipedia page, how can I characterize the above defined matrices?