Matrix normal distribution

Matrix normal
Matrix normal
Notation
Parameters	location (real matrix); scale (positive-definite real matrix); scale (positive-definite real matrix)
Support
PDF
Mean
Variance	(among-row) and (among-column)

In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.

Matrix normal
Notation	${\mathcal {MN}}_{n,p}(\mathbf {M} ,\mathbf {U} ,\mathbf {V} )$
Parameters	$\mathbf {M}$ location (real $n\times p$ matrix) $\mathbf {U}$ scale (positive-definite real $n\times n$ matrix) $\mathbf {V}$ scale (positive-definite real $p\times p$ matrix)
Support	$\mathbf {X} \in \mathbb {R} ^{n\times p}$
PDF	${\frac {\exp \left(-{\frac {1}{2}}\,\mathrm {tr} \left[\mathbf {V} ^{-1}(\mathbf {X} -\mathbf {M} )^{T}\mathbf {U} ^{-1}(\mathbf {X} -\mathbf {M} )\right]\right)}{(2\pi )^{np/2}\|\mathbf {V} \|^{n/2}\|\mathbf {U} \|^{p/2}}}$
Mean	$\mathbf {M}$
Variance	$\mathbf {U}$ (among-row) and $\mathbf {V}$ (among-column)