Let \mu and \nu be two probability measures on the interval [0,+\infty), and assume that X is a random variable in a non commutative probability space with law \mu and Y is a random variable in the same non commutative probability space with law \nu. Assume finally that X and Y are freely independent. Then the free multiplicative convolution \mu\boxtimes\nu is the law of X^{1/2}YX^{1/2} (or, equivalently, the law of Y^{1/2}XY^{1/2}. Random matrices interpretation: if A and B are some independent n by n non negative Hermitian (resp. real symmetric) random matrices such that at least one of them is invariant, in law, under conjugation by any unitary (resp. orthogonal) matrix and such that the empirical spectral measures of A and B tend respectively to \mu and \nu as n tends to infinity, then the empirical spectral measure of AB tends to \mu\boxtimes\nu.Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010). An introduction to random matrices. Cambridge: Cambridge University Press. .
