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.

The probability density function for the random matrix X (n × p) that follows the matrix normal distribution ${\displaystyle {\mathcal {MN}}_{n,p}(\mathbf {M} ,\mathbf {U} ,\mathbf {V} )}$ has the form:

where ${\displaystyle \mathrm {tr} }$ 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 ${\displaystyle \mathbb {R} ^{n\times p}}$, i.e.: the measure corresponding to integration with respect to ${\displaystyle dx_{11}dx_{21}\dots dx_{n1}dx_{12}\dots dx_{n2}\dots dx_{np}}$.

The matrix normal is related to the multivariate normal distribution in the following way:

if and only if

where ${\displaystyle \otimes }$ denotes the Kronecker product and ${\displaystyle \mathrm {vec} (\mathbf {M} )}$ denotes the vectorization of ${\displaystyle \mathbf {M} }$.

The equivalence between the above matrix normal and multivariate normal density functions can be shown using several properties of the trace and Kronecker product, as follows. We start with the argument of the exponent of the matrix normal PDF:

which is the argument of the exponent of the multivariate normal PDF with respect to Lebesgue measure in ${\displaystyle \mathbb {R} ^{np}}$. The proof is completed by using the determinant property: ${\displaystyle |\mathbf {V} \otimes \mathbf {U} |=|\mathbf {V} |^{n}|\mathbf {U} |^{p}.}$