Functions of singular random matrices with applications

This article describes how the Jacobian is found for certain functions of a singular random matrix, both in the general case and in that of a non-negative definite random matrix. The Jacobian of the transformation V = S-2 is found when S is non-negative definite; in addition, the Jacobian of the transformation Y = X+ is determined when X+ is the generalized, or Moore-Penrose, inverse of X. Expressions for the densities of the generalized inverse of the central beta and F singular random matrices are proposed. Finally, two applications in the field of Bayesian inference are presented.