We present the loop space optimization framework for solving large $N$ multi-matrix systems. Based on the collective, loop space representations, the scheme provides a systematic and efficient numerical approach for solving both multi-matrix integrals ($c=0$ systems) and also multi-matrix quantum mechanics ($c=1$ systems). Positivity constraints are of significant importance and are addressed through master-field minimization. The framework not only facilitates the determination of large $N$ backgrounds, but also enables a systematic $1/N$ expansion. Consequently the complete fluctuation spectrum is also computable, which holds immediate physical relevance in matrix quantum mechanics. The complexity and the rapid growth of the degrees of freedom for these theories have stymied earlier attempts, and in this thesis we present significant improvement in this regard. To illustrate the effectiveness of our approach, we apply this scheme to two-matrix integrals and two-matrix quantum mechanics. These examples involve minimization and spectrum calculations with close to $10^4$ variables, providing solutions to the Migdal-Makeenko (loop) and collective field equations. Despite the large number of dynamical loop variables and the extreme nonlinearity of the problem, our scheme achieves high precision when confronted with solvable cases, demonstrating its capability to solve general two-matrix problems. Additionally, we present preliminary results for the matrix thermofield double. We also briefly discuss other available numerical methods, namely the Monte Carlo method and Matrix Bootstrap.
