In this thesis, we present a new methodology to investigate the existence and uniqueness of the solution of the Social Certainty Equivalence (SCE) equation system related to social optima in mean field linear-quadratic-Gaussian (LQG) control problems. The methodology involves Hamiltonian matrices and continuous-time algebraic Riccati equations (CARE), and it avoids the restrictive contractive assumption typically used in a fixed point approach. For computing the stabilizing solution of CARE, we also develop a computational method related to generalized eigenvectors and Schur vectors. We further extend our method to solve a linear-quadratic mean field game.

Keywords: SCE Equation System; Hamiltonian Matrices; Continuous-time Algebraic Riccati Equations