Notation ${\rm {T))_{n,p}(\nu ,\mathbf {M} ,{\boldsymbol {\Sigma )),{\boldsymbol {\Omega )))$ $\mathbf {M}$ location (real $n\times p$ matrix) ${\boldsymbol {\Omega ))$ scale (positive-definite real $p\times p$ matrix) ${\boldsymbol {\Sigma ))$ scale (positive-definite real $n\times n$ matrix) $\nu$ degrees of freedom $\mathbf {X} \in \mathbb {R} ^{n\times p)$ ${\frac {\Gamma _{p}\left({\frac {\nu +n+p-1}{2))\right)}{(\pi )^{\frac {np}{2))\Gamma _{p}\left({\frac {\nu +p-1}{2))\right)))|{\boldsymbol {\Omega ))|^{-{\frac {n}{2))}|{\boldsymbol {\Sigma ))|^{-{\frac {p}{2)))$ $\times \left|\mathbf {I} _{n}+{\boldsymbol {\Sigma ))^{-1}(\mathbf {X} -\mathbf {M} ){\boldsymbol {\Omega ))^{-1}(\mathbf {X} -\mathbf {M} )^{\rm {T))\right|^{-{\frac {\nu +n+p-1}{2)))$ No analytic expression $\mathbf {M}$ if $\nu +p-n>1$ , else undefined $\mathbf {M}$ ${\frac ((\boldsymbol {\Sigma ))\otimes {\boldsymbol {\Omega ))}{\nu -2))$ if $\nu >2$ , else undefined see below

In statistics, the matrix t-distribution (or matrix variate t-distribution) is the generalization of the multivariate t-distribution from vectors to matrices. The matrix t-distribution shares the same relationship with the multivariate t-distribution that the matrix normal distribution shares with the multivariate normal distribution.[clarification needed] For example, the matrix t-distribution is the compound distribution that results from sampling from a matrix normal distribution having sampled the covariance matrix of the matrix normal from an inverse Wishart distribution.[citation needed]

In a Bayesian analysis of a multivariate linear regression model based on the matrix normal distribution, the matrix t-distribution is the posterior predictive distribution.

## Definition

For a matrix t-distribution, the probability density function at the point $\mathbf {X}$ of an $n\times p$ space is

$f(\mathbf {X} ;\nu ,\mathbf {M} ,{\boldsymbol {\Sigma )),{\boldsymbol {\Omega )))=K\times \left|\mathbf {I} _{n}+{\boldsymbol {\Sigma ))^{-1}(\mathbf {X} -\mathbf {M} ){\boldsymbol {\Omega ))^{-1}(\mathbf {X} -\mathbf {M} )^{\rm {T))\right|^{-{\frac {\nu +n+p-1}{2))},$ where the constant of integration K is given by

$K={\frac {\Gamma _{p}\left({\frac {\nu +n+p-1}{2))\right)}{(\pi )^{\frac {np}{2))\Gamma _{p}\left({\frac {\nu +p-1}{2))\right)))|{\boldsymbol {\Omega ))|^{-{\frac {n}{2))}|{\boldsymbol {\Sigma ))|^{-{\frac {p}{2))}.$ Here $\Gamma _{p)$ is the multivariate gamma function.

The characteristic function and various other properties can be derived from the generalized matrix t-distribution (see below).

## Generalized matrix t-distribution

Notation ${\rm {T))_{n,p}(\alpha ,\beta ,\mathbf {M} ,{\boldsymbol {\Sigma )),{\boldsymbol {\Omega )))$ $\mathbf {M}$ location (real $n\times p$ matrix) ${\boldsymbol {\Omega ))$ scale (positive-definite real $p\times p$ matrix) ${\boldsymbol {\Sigma ))$ scale (positive-definite real $n\times n$ matrix) $\alpha >(p-1)/2$ shape parameter $\beta >0$ scale parameter $\mathbf {X} \in \mathbb {R} ^{n\times p)$ ${\frac {\Gamma _{p}(\alpha +n/2)}{(2\pi /\beta )^{\frac {np}{2))\Gamma _{p}(\alpha )))|{\boldsymbol {\Omega ))|^{-{\frac {n}{2))}|{\boldsymbol {\Sigma ))|^{-{\frac {p}{2)))$ $\times \left|\mathbf {I} _{n}+{\frac {\beta }{2)){\boldsymbol {\Sigma ))^{-1}(\mathbf {X} -\mathbf {M} ){\boldsymbol {\Omega ))^{-1}(\mathbf {X} -\mathbf {M} )^{\rm {T))\right|^{-(\alpha +n/2))$ $\Gamma _{p)$ is the multivariate gamma function. No analytic expression $\mathbf {M}$ ${\frac {2({\boldsymbol {\Sigma ))\otimes {\boldsymbol {\Omega )))}{\beta (2\alpha -p-1)))$ see below

The generalized matrix t-distribution is a generalization of the matrix t-distribution with two parameters α and β in place of ν.

This reduces to the standard matrix t-distribution with $\beta =2,\alpha ={\frac {\nu +p-1}{2)).$ The generalized matrix t-distribution is the compound distribution that results from an infinite mixture of a matrix normal distribution with an inverse multivariate gamma distribution placed over either of its covariance matrices.

### Properties

If $\mathbf {X} \sim {\rm {T))_{n,p}(\alpha ,\beta ,\mathbf {M} ,{\boldsymbol {\Sigma )),{\boldsymbol {\Omega )))$ then[citation needed]

$\mathbf {X} ^{\rm {T))\sim {\rm {T))_{p,n}(\alpha ,\beta ,\mathbf {M} ^{\rm {T)),{\boldsymbol {\Omega )),{\boldsymbol {\Sigma ))).$ The property above comes from Sylvester's determinant theorem:

$\det \left(\mathbf {I} _{n}+{\frac {\beta }{2)){\boldsymbol {\Sigma ))^{-1}(\mathbf {X} -\mathbf {M} ){\boldsymbol {\Omega ))^{-1}(\mathbf {X} -\mathbf {M} )^{\rm {T))\right)=$ $\det \left(\mathbf {I} _{p}+{\frac {\beta }{2)){\boldsymbol {\Omega ))^{-1}(\mathbf {X} ^{\rm {T))-\mathbf {M} ^{\rm {T))){\boldsymbol {\Sigma ))^{-1}(\mathbf {X} ^{\rm {T))-\mathbf {M} ^{\rm {T)))^{\rm {T))\right).$ If $\mathbf {X} \sim {\rm {T))_{n,p}(\alpha ,\beta ,\mathbf {M} ,{\boldsymbol {\Sigma )),{\boldsymbol {\Omega )))$ and $\mathbf {A} (n\times n)$ and $\mathbf {B} (p\times p)$ are nonsingular matrices then[citation needed]

$\mathbf {AXB} \sim {\rm {T))_{n,p}(\alpha ,\beta ,\mathbf {AMB} ,\mathbf {A} {\boldsymbol {\Sigma ))\mathbf {A} ^{\rm {T)),\mathbf {B} ^{\rm {T)){\boldsymbol {\Omega ))\mathbf {B} ).$ $\phi _{T}(\mathbf {Z} )={\frac {\exp({\rm {tr))(i\mathbf {Z} '\mathbf {M} ))|{\boldsymbol {\Omega ))|^{\alpha )){\Gamma _{p}(\alpha )(2\beta )^{\alpha p))}|\mathbf {Z} '{\boldsymbol {\Sigma ))\mathbf {Z} |^{\alpha }B_{\alpha }\left({\frac {1}{2\beta ))\mathbf {Z} '{\boldsymbol {\Sigma ))\mathbf {Z} {\boldsymbol {\Omega ))\right),$ where

$B_{\delta }(\mathbf {WZ} )=|\mathbf {W} |^{-\delta }\int _{\mathbf {S} >0}\exp \left({\rm {tr))(-\mathbf {SW} -\mathbf {S^{-1}Z} )\right)|\mathbf {S} |^{-\delta -{\frac {1}{2))(p+1)}d\mathbf {S} ,$ and where $B_{\delta )$ is the type-two Bessel function of Herz[clarification needed] of a matrix argument.