In statistics, the matrix variate beta distribution is a generalization of the beta distribution.

If

p × p

positive definite matrix with a matrix variate beta distribution, and

are real parameters, we write

)

The probability density function for

β

is the multivariate beta function: where

is the multivariate gamma function given by If

then the density of

is given by provided that

is a constant

orthogonal matrix, then

is a random orthogonal

matrix which is independent of

, distributed independently of

is any constant

matrix of rank

has a generalized matrix variate beta distribution, specifically

and we partition

, then defining the Schur complement

gives the following 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

has a matrix variate beta distribution

is independent of