A function $f : \mathbb{R}^{n} \rightarrow \mathbb{R}^n$ is monotone with respect to $P = P^\top\succcurlyeq 0$ if $$ \left( f(x) - f(y) \right)^\top P (x-y) \geq 0 $$ for all $x,y$.
Now suppose that $g : \mathbb{R}^{n} \rightarrow \mathbb{R}^n$ is such that $$ \left( g(x) - g(y) \right)^\top A (x-y) \geq 0$$ for all $x,y$, where $0 \neq A \neq A^\top$.
Is there some $0 \neq Q = Q^\top \succcurlyeq 0$ such that $g$ is monotone with respect to $Q$?
Comment. I tried to consider $A + A^\top$ as candidate $Q$ but it did not worked out.