We consider a nonlinear system of ODEs, where the underlying linear dynamics
are determined by a Hermitian random matrix ensemble. We prove that the leading
order dynamics in the weakly nonlinear, infinite volume limit are determined by
a solution to the corresponding kinetic wave equation on a non-trivial
timescale. Our proof relies on estimates for Haar-distributed unitary matrices
obtained from Weingarten calculus, which may be of independent interest.