I would like to prove this statement:
"$F$ is monotone if and only if $\nabla F$ is positive semidefinite."
I only know $F$ is monotone with respect to $\Omega$ if and only if
$$(u-v)^{T}(F(u)-F(v)) \geq 0, \forall u, v \in \Omega.$$
Should I start with $x^{T} \nabla F x$ and then arrive at this stuff is larger or equal to 0?
Thank you in advance.
P.S. $F$ is assumed to be differentiable in the context.
It depends what you mean by "positive semidefinite". Some people require the matrix to be symmetric others don't. The following is true:
Theorem. $F$ is monotone if and only if the symmetric part of the Jacobian $\nabla F(x)$ is positive semidefinite for all $x$.
An example could be the rotator by $\pi/2$ in the Euclidean plane. This is a skew operator with symmetric part being the zero matrix.
For a proof of the characterization theorem, see the book Variational Analysis by Rockafellar and Wets, Proposition 12.3.