The Law of Multiplication of Large Random Matrices Revisited

The paper deals with the eigenvalue distribution of the product of two n x n positive definite matrices B tau, tau = +/- 1, rotated with respect to each other by the random orthogonal and Haar distributed matrix. The prob-lem has been considered in several works by using various techniques. We propose a streamlined approach based on the random matrix theory tech-niques and a certain symmetry of the problem. We prove the convergence with probability 1 as n tends to infinity of the Normalized Counting Mea-sure (NCM) of eigenvalues of the product to a non-random limit, derive a functional equation that determines the Stieltjes transform of the limiting NCM of the product in terms of limiting NCMs of the factors B tau, tau = +/- 1, and consider an interesting example.