Sample from matrix normal 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:

Find the probability that smallest eigenvalue :

Probability density function:

Plot PDF for a diagonal matrices:

Skewness and kurtosis:

Visualize sample matrices from matrix normal distributions:

Use matrix normal distribution to simulate a vector autoregressive process:

Construct TemporalData from sampled values:

Estimate diagonal vector autoregressive process:

Compare to the original values:

Matrix normal 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:

Sample from matrix normal distribution with independent rows:

Test the hypothesis that rows follow multinormal distribution with the column covariance matrix:

Sample from matrix normal distribution with independent rows:

Test the hypothesis that rows follow multinormal distribution with the column covariance matrix:

Sample from matrix normal distribution with independent rows:

Computing sample inter-row covariances shows different rows are pairwise independent:

Computing sample inter-column covariances shows different columns are dependent:

By joining the rows of the matrix-valued random variable together, a matrix normal distribution can be regarded as a multivariate normal distribution:

The covariance matrix of the vectorized random matrix is the Kronecker product of and :

Matrix normal 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 normal 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 distribution indicates that the estimate is good: