Matrix normal distribution

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. == Definition == The probability density function for the random matrix X (n × p) that follows the matrix normal distribution M N n , p ( M , U , V ) {\displaystyle {\mathcal {MN}}_{n,p}(\mathbf {M} ,\mathbf {U} ,\mathbf {V} )} has the form: p ( X ∣ M , U , V ) = exp ⁡ ( − 1 2 t r [ V − 1 ( X − M ) T U − 1 ( X − M ) ] ) ( 2 π ) n p / 2 | V | n / 2 | U | p / 2 {\displaystyle p(\mathbf {X} \mid \mathbf {M} ,\mathbf {U} ,\mathbf {V} )={\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}}}} where t r {\displaystyle \mathrm {tr} } denotes trace and M is n × p, U is n × n and V is p × p, and the density is understood as the probability density function with respect to the standard Lebesgue measure in R n × p {\displaystyle \mathbb {R} ^{n\times p}} , i.e.: the measure corresponding to integration with respect to d x 11 d x 21 … d x n 1 d x 12 … d x n 2 … d x n p {\displaystyle dx_{11}dx_{21}\dots dx_{n1}dx_{12}\dots dx_{n2}\dots dx_{np}} .

Source: Wikipedia — Matrix normal distribution (CC BY-SA 4.0)