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(tr AA†), 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 = B ^-1X. 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). The universal probability distribution P(Z) is a spherically symmetric matrix-variate t-distribution. Universality of P(Z) is, essentially, a consequence of rotational invariance of the probability ensembles we study. 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.