The Matsumoto and Yor obeserved that that if X and Y are independent random variables, such that X has a Generalized Inverse Gaussian (GIG) distribution and Y has a Gamma distribution then $U=(X+Y)^{-1}$ and $V=X^{-1}-(X+Y)^{-1}$ are independent. 

Later G. Letac and J. Wesolowski proved that the GIG and the Gamma are the only distributions which have the above property and proved analogous results for Wishart and GIG matrices.

It turns out that a similar result holds in non-commutative probability.

The talk will contain presentation of all necessary notions and results from free probability.