This article presents an analysis of selected characteristics of the Modular Multilevel Matrix Converter (M3C) operating a doubly fed induction generator (DFIG) in a flywheel energy storage system (FESS). A detailed analysis of the necessary electrical input and output quantities, as well as the identification of required internal currents and voltages, leads to a newly introduced iterative design process for the converter. This process ultimately provides information on the required number of submodules and demonstrates the overload capability of the converter in reactive power operation. Also the short circuit current contribution of the system for this unique configuration of M3C and DFIG is investigated for the first time. Taking into account the unique operating characteristics of the M3C, a loss analysis of the converter and the machine is performed, and a loss-optimized distribution of reactive power between the machine and the converter is proposed. The study is conducted analytically and is validated by experimental results on a small-scale demonstrator.