On the universality of the probability distribution of the product $B^{-1}X$ of random matrices

Joshua Feinberg

Published 2002-04-25, updated 2004-02-21Version 2

Consider random matrices $A$, of dimension $m\times (m+n)$, drawn from an ensemble with probability density $f(\rmtr AA^\dagger)$, with $f(x)$ a given appropriate function. Break $A = (B,X)$ into an $m\times m$ block $B$ and the complementary $m\times n$ block $X$, and define the random matrix $Z=B^{-1}X$. 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.