Pseudo-inverse multivariate/matrix-variate distributions

The Moore–Penrose inverse of a singular or nonsquare matrix is not only existent but also unique. In this paper, we derive the Jacobian of the transformation from such a matrix to the transpose of its Moore–Penrose inverse. Using this Jacobian, we investigate the distribution of the Moore–Penrose inverse of a random matrix and propose the notion of pseudo-inverse multivariate/matrix-variate distributions. For arbitrary multivariate or matrix-variate distributions, we can develop the corresponding pseudo-inverse distributions. In particular, we present pseudo-inverse multivariate normal distributions, pseudo-inverse Dirichlet distributions, pseudo-inverse matrix-variate normal distributions and pseudo-inverse Wishart distributions.