Distribution of the generalised inverse of a random matrix and its applications

Given a random singular matrix X  , in the present article we find the Jacobian of the transformation Y=X+Y=X+, where X+X+ is the Moore–Penrose inverse of X, both in the general case and when X is a non-negative definite matrix. Expressions for the densities of the Moore–Penrose inverse of the singular Wishart and Pseudo-Wishart matrices are obtained. Similarly, an expression for the density of the matrix-variate singular T-distribution is proposed. Finally, these results are applied to the Bayesian inference of the multivariate linear model.