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:

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

If ${\displaystyle U\sim B_{p}(a,b)}$ then the density of ${\displaystyle X=U^{-1}}$ is given by

provided that ${\displaystyle X>I_{p}}$ and ${\displaystyle a,b>(p-1)/2}$.

If ${\displaystyle U\sim B_{p}(a,b)}$ and ${\displaystyle H}$ is a constant ${\displaystyle p\times p}$ orthogonal matrix, then ${\displaystyle HUH^{T}\sim B(a,b).}$

Also, if ${\displaystyle H}$ is a random orthogonal ${\displaystyle p\times p}$ matrix which is independent of ${\displaystyle U}$, then ${\displaystyle HUH^{T}\sim B_{p}(a,b)}$, distributed independently of ${\displaystyle H}$.