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.)



Free multiplicative convolution of two random matrices
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](https://rolandspeicher.com/literature/mingo-speicher/)
or the article
[Helton, Mai, Speicher: Applications of Realizations (aka Linearizations) to Free Probability](https://arxiv.org/pdf/1511.05330.pdf)


