Matrix variate beta distribution

Matrix variate beta distribution
Notation	${\displaystyle {\rm {B}}_{p}(a,b)}$
Parameters	${\displaystyle a,b}$
Support	${\displaystyle p\times p}$ matrices with both ${\displaystyle U}$ and ${\displaystyle I_{p}-U}$ positive definite
PDF	${\displaystyle \left\{\beta _{p}\left(a,b\right)\right\}^{-1}\det \left(U\right)^{a-(p+1)/2}\det \left(I_{p}-U\right)^{b-(p+1)/2}.}$
CDF	${\displaystyle {}_{1}F_{1}\left(a;a+b;iZ\right)}$

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 ${\displaystyle U}$ is a ${\displaystyle p\times p}$ positive definite matrix with a matrix variate beta distribution, and ${\displaystyle a,b>(p-1)/2}$ are real parameters, we write ${\displaystyle U\sim B_{p}\left(a,b\right)}$ (sometimes ${\displaystyle B_{p}^{I}\left(a,b\right)}$). The probability density function for ${\displaystyle U}$ is:$${\displaystyle \left\{\beta _{p}\left(a,b\right)\right\}^{-1}\det \left(U\right)^{a-(p+1)/2}\det \left(I_{p}-U\right)^{b-(p+1)/2}.}$$

Here ${\displaystyle \beta _{p}\left(a,b\right)}$ is the multivariate beta function:

${\displaystyle \beta _{p}\left(a,b\right)={\frac {\Gamma _{p}\left(a\right)\Gamma _{p}\left(b\right)}{\Gamma _{p}\left(a+b\right)}}}$

where ${\displaystyle \Gamma _{p}\left(a\right)}$ is the multivariate gamma function given by

${\displaystyle \Gamma _{p}\left(a\right)=\pi ^{p(p-1)/4}\prod _{i=1}^{p}\Gamma \left(a-(i-1)/2\right).}$