This paper deals with a control method and a designing method of the capacitance of the compensation capacitor for a single phase to three phase matrix converters (MCs) with the variable speed drive capability of induction motors (IMs). Assuming the application of railway tractions, automobiles and elevators, motors drive inertial load. In such cases, both the load power and the load voltage increase in proportion to the rotor speed. The amplitude of the compensation capacitor voltage is controlled to absorb the single phase power fluctuation, along with the load power. We propose a method to decide the input side parameters such as the capacitance of the compensation capacitor, considering the input voltage and the power of the IM. We experimentally verify that a single phase to three phase MC can drive IMs with the proposed method.