This paper extends the study of Wishart and multivariate beta distributions to the singular case, where the rank is below the dimensionality. The usual conjugacy is extended to this case. A volume element on the space of positive semidefinite $m \times m$ matrices of rank $n < m$ is introduced and some transformation properties established. The density function is found for all rank-$n$ Wishart distributions as well as the rank-1 multivariate beta distribution. To do that, the Jacobian for the transformation to the singular value decomposition of general $m \times n$ matrices is calculated. The results in this paper are useful in particular for updating a Bayesian posterior when tracking a time-varying variance-covariance matrix.
"On Singular Wishart and Singular Multivariate Beta Distributions." Ann. Statist. 22 (1) 395 - 405, March, 1994. https://doi.org/10.1214/aos/1176325375