On the universality of the probability distribution of the product B−1X of random matrices

Consider random matrices A, of dimension m× (m+ n), drawn from an ensemble with probability density f(trAA), with f(x) a given appropriate function. Break A = (B,X) into an m×m block B and the complementary m× n block X , and define the random matrix Z = BX . We calculate the probability density function P (Z) of the random matrix Z and find that it is a universal function, independent of f(x). Universality of P (Z) is, essentially, a consequence of rotational invariance of the probability ensembles we study. More generally, P (Z) must be independent, of course, of any common scale of the distribution functions of B and X . As an application, we study the distribution of solutions of systems of linear equations with random coefficients, and extend a classic result due to Girko. *e-mail address: [email protected] # permanent address