# MatrixNormalDistribution

MatrixNormalDistribution[Σrow,Σcol]

represents zero mean matrix normal distribution with row covariance matrix Σrow and column covariance matrix Σcol.

MatrixNormalDistribution[μ,Σrow,Σcol]

represents matrix normal distribution with mean matrix μ.

# Details • MatrixNormalDistribution is a distribution of μ+ .x. , where is a matrix with independent identically distributed matrix elements that follow NormalDistribution[0,1].
• The probability density for a matrix in a matrix normal distribution is proportional to .
• MatrixNormalDistribution[μ,c Σrow,c-1 Σcol] has the same distribution as MatrixNormalDistribution[μ,Σrow,Σcol] for any positive real constant c.
• The covariance matrices Σrow and Σcol can be any symmetric positive definite matrices of real numbers of dimensions {n,n} and {m,m}, respectively, and the mean matrix μ can be any matrix of real numbers of dimensions {n,m}.
• MatrixNormalDistribution can be used with such functions as MatrixPropertyDistribution, EstimatedDistribution, and RandomVariate.

# Examples

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## Basic Examples(2)

Sample from matrix normal distribution:

Mean and variance:

## Scope(7)

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:

## Applications(2)

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:

## Properties & Relations(6)

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[Σrow,Σcol,ν] is a parameter mixture of MatrixNormalDistribution[Σ,Σcol] with following InverseWishartMatrixDistribution[ν+n-1,Σrow]:
• 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 :

## Possible Issues(1)

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: