Given two freely independent random hermitian matrices $A$ and $B$ following laws $\mu, \nu$, one can compute the empirical spectral distribution of $AB$ by their free multiplicative convolution $\mu\boxtimes\nu$ using the $S$-transform. Is there a way to compute the empirical spectral distribution of other products of $A$ and $B$, such as $AB^{-1}$?
(I am new to random matrix theory so the question might sound naive.)
Best Answer
There is the possibility of dealing with any non-commutative rational function in A and B, by using more general (operator-valued) versions of free probability. Usually the corresponding equations cannot be solved analytically, but they are nice fixed-point equations which can be addressed numerically. See for example, chapter 10 in the book Mingo, Speicher: Free Probability and Random Matrices or the article Helton, Mai, Speicher: Applications of Realizations (aka Linearizations) to Free Probability