  The Lohe matrix model is a continuous-time dynamical system describing the
collective dynamics of group elements in the unitary group manifold, and it has
been introduced as a toy model of a non abelian generalization of the Kuramoto
phase model. In the absence of couplings, it reduces to the finite-dimensional
decoupled free Schr\"{o}dinger equations with constant Hamiltonians. In this
paper, we study a rigorous mean-field limit of the Lohe matrix model which
results in a Vlasov type equation for the probability density function on the
corresponding phase space. We also provide two different settings for the
emergent synchronous dynamics of the kinetic Lohe equation in terms of the
initial data and the coupling strength.
