I'm not quite sure how to interpret your term "matrix converter". My best guest would be a "motor inverter" (3-phase?) working from the grid.
To answer your question in this light: though you can virtually do anything with any microcontroller, this is not completely true. Though the RPi sports impressing computing power, it lacks nearly all support for the necessary hardware interfaces.
The arduino hardware could do, but with considerable problems. And not with the "standard" arduino software environment. Everything just too slow.
When considering inverter applications I'd suggest to look for a microcontroller with at least 6 PWM outputs with "deadtime" support and fast ADCs (best with >= 2 individual ADC units).
Looking at microchip.com or atmel.com - especially for application notes about inverter problems - should give you some hints about how such microcontrollers "look like". Applicable controllers are offered by nearly all microcontroller vendors, but I'd suggest Microchip or Atmel for ease of application and free resp. low-cost development tools.
I do not think that Arduino will do the job, because, for 3 phase 3 legs matrix converter you you have 9 IGBTs and each has 2 switches, so you need 18 I/O line.
Also, with switching frequency 10KHz for example, you need 100 micro second to do the ( ADC read+ filters+ modulation+ control+send data to I/O ) and i donot think you can manage to do that in Arduino.
In my case, i have used DSP from TI, to do the modulation and calculation of control algorithm than store the results of duty cycles in shared memory between DSP and FPGA, and the FPGA will feach the Duty cycles from the shared memory and send them to IGBT gate drive pins.
you must do hybride control if you want increase the switching frequency by using DSP or Dspace to manage the input and to generate duty cycle then the arduino generates the signals and also here you have a problem which is how to configure the PWM module of the arduino to generate hardware PWM signals (i think it is possible but very difficult)