Sample from matrix distribution:

Mean and variance:

Generate a single pseudorandom matrix:

Generate a single pseudorandom matrix with nonzero mean:

Generate a set of pseudorandom matrices:

Sample at extended precision:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare LogLikelihood for both distributions:

Skewness and kurtosis:

Probability density function:

Plot PDF for a diagonal matrices:

Matrix t distribution is defined up to a positive multiplicative constant:

Equivalent distribution with row and column scale matrices multiplied and divided by a positive constant:

Compute the PDF of the distributions at a random point:

MatrixTDistribution[Σ_row,Σ_col,ν] is a parameter mixture of MatrixNormalDistribution[Σ,Σ_col] with following InverseWishartMatrixDistribution[ν+n-1,Σ_row]:

Create a sample following the parameter mixture of MatrixNormalDistribution with InverseWishartMatrixDistribution:

Fit the sample data to MatrixTDistribution:

Compute log-likelihood ratio statistic against the appropriate MatrixTDistribution

Log-likelihood ratio follows ChiSquareDistribution with the parameter equal to the number of degrees of freedom:

Compute the -value of log-likelihood ratio test:

For matrix sampled from matrix distribution, the expression follows Student distribution for any nonzero vectors and with lengths that match with the dimension of :

Use MatrixPropertyDistribution to sample values of the expression :

Check agreement with the expected distribution:

For matrix sampled from matrix distribution, follows multivariate distribution for any nonzero vector with length that matches with the number of columns of :

Use MatrixPropertyDistribution to sample values of :

Verify goodness of fit with the expected distribution:

Matrix distribution is defined up to a multiplicative scaling constant. The estimated parameters may not be close to the ones that specify the underlying distribution:

Sample from the matrix distribution:

Estimate the distribution:

Compare the estimated scale parameters with the ones of the underlying distribution:

Kronecker products of the scale matrices are close to each other:

The LogLikelihood of the distributions indicate that the estimate is good: