On singular matrix variate beta distribution

In this paper the density of the matrix variate beta distribution of rank lower than its dimensionality is obtained with respect to a suitably defined differential form under the condition that the difference between the identity and this matrix has full rank. As preliminaries, the Jacobian of a transformation related to decomposing a nonnegative-definite matrix into the product of a matrix of full column rank and its transpose and that of the transformation of a nonnegative-definite matrix into its congruent matrix are established.