These are just DC brushless motors with 3 windings in the body of the motor, and permanent magnets on the shaft. Each winding is driven by a PWM signal, and the PWM signals are out of phase, i.e. one signals leads the next signal which leads the next signal. This causes a rotating magnetic field around the core, which pushes on the magnets of the shaft and cause it to rotate. The speed and direction of rotation are controlled by the pwm and the order in which the windings are energized.
The phase matrix is simply a 3-phase full-bridge controller consisting of 6 MOSFETs. This lets the controller provide acceleration torque and deceleration torque in both forward and reverse directions. This is known as 4 quadrant operation.
There are many different algorithms for controlling the 6 pwm signals needed. You should read up on 3-phase DC motor control theory and about the PWM algorithms in particular. The algorithms cover the wave shapes, the transition times, PWM frequencies, etc... everything needed to get the motor going in the direction you want at the speed you want with the amount of torque needed.
Whether or not you can control this with the Arduino depends on the arduino having a 3-phase full bridge and 6 available pwm outputs. Or a shield for that. I don't know anything Arduino so I can't help there.