We describe a method for unmixing mixtures of 'freely' independent random variables in a manner analogous to the indepedent component analysis (ICA) based method for unmixing independent random variables from their additive mixture. Random matrices play the role of free random variables in this context so the method we develop, which we call Free component analysis (FCA), unmixes matrices from an additive mixture of matrices. We describe the theory -- the various 'contrast functions', computational methods and compare FCA to ICA on data derived from real-world experiments.