Matrix variate beta distribution

In statistics, the matrix variate beta distribution is a generalization of the beta distribution. It is also called the MANOVA ensemble and the Jacobi ensemble.
If is a positive definite matrix with a matrix variate beta distribution, and are real parameters, we write . The probability [density function] for is:
Here is the multivariate beta function:
where is the multivariate gamma function given by

Theorems

Distribution of matrix inverse

If then the density of is given by
provided that and.

Orthogonal transform

If and is a constant orthogonal matrix, then
Also, if is a random orthogonal matrix which is independent of, then, distributed independently of.
If is any constant, matrix of rank, then has a generalized matrix variate beta distribution, specifically.

Partitioned matrix results

If and we partition as
where is and is, then defining the Schur complement as gives the following results:

is independent of
has an inverted matrix variate t distribution, specifically

Wishart results

Mitra proves the following theorem which illustrates a useful property of the matrix variate beta distribution. Suppose are independent Wishart matrices. Assume that is positive definite and that. If
where, then has a matrix variate beta distribution. In particular, is independent of.

Spectral density

The spectral density is expressed by a Jacobi polynomial.

Extreme value distribution

The distribution of the largest eigenvalue is well approximated by a transform of the Tracy–Widom distribution.