  This paper studies the existence and uniqueness of a solution to linear
quadratic (LQ) mean field social optimization problems with uniform agents. We
exploit a Hamiltonian matrix structure of the associated ordinary differential
equation (ODE) system and apply a subspace decomposition method to find the
solution.
  This approach is effective for both the existence analysis and numerical
computations.
  We further extend the decomposition method to LQ mean field games.
