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 transformationV=S ² is found whenS is non-negative definite; in addition, the Jacobian of the transformationY=X ⁺ is determined whenX ⁺ is the generalized, or Moore-Penrose, inverse ofX. 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. This work was supported in part by the research project 39017E of CONACYT-México This article was written during the first author's stay as a Visiting Professor in the Department of Statistics at the University of Granada, Spain