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: If the matrix has only one row, or only one column, the distributions become equivalent to the corresponding (vector-)multivariate distribution. The matrix t-distribution is the compound distribution that results from an infinite mixture of a matrix normal distribution with an inverse Wishart distribution placed over either of its covariance matrices, and the multivariate t-distribution can be generated in a similar way.

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.

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

where the constant of integration K is given by

Here ${\displaystyle \Gamma _{p}}$ is the multivariate gamma function.

If ${\displaystyle \mathbf {X} \sim {\mathcal {T}}_{n\times p}(\nu ,\mathbf {M} ,\mathbf {\Sigma } ,\mathbf {\Omega } )}$, then we have the following properties:

The mean, or expected value is, if ${\displaystyle \nu >1}$: