Matrix normal distribution

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)

In statistics, the matrix normal 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 $\mathcal{MN}_{n,p}(\mathbf{M}, \mathbf{U}, \mathbf{V})$ has the form:

$p(\mathbf{X}|\mathbf{M}, \mathbf{U}, \mathbf{V}) = \frac{\exp\left( -\frac{1}{2} \, \mathrm{tr}\left[ \mathbf{V}^{-1} (\mathbf{X} - \mathbf{M})^{T} \mathbf{U}^{-1} (\mathbf{X} - \mathbf{M}) \right] \right)}{(2\pi)^{np/2} |\mathbf{V}|^{n/2} |\mathbf{U}|^{p/2}}$

where M is n × p, U is n × n and V is p × p.

There are several ways to define the two covariance matrices. One possibility is

$\mathbf{U} = E[(\mathbf{X} - \mathbf{M})(\mathbf{X} - \mathbf{M})^{T}]$

$\mathbf{V} = E[(\mathbf{X} - \mathbf{M})^{T} (\mathbf{X} - \mathbf{M})] / c$

where $c$ is a constant which depends on U and ensures appropriate power normalization.

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

$\mathbf{X} \sim \mathcal{MN}_{n\times p}(\mathbf{M}, \mathbf{U}, \mathbf{V}),$

if and only if

$\mathrm{vec}(\mathbf{X}) \sim \mathcal{N}_{np}(\mathrm{vec}(\mathbf{M}), \mathbf{V} \otimes \mathbf{U})$

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

Example

Let's imagine a sample of n independent p-dimensional random variables identically distributed according to a multivariate normal distribution:

$\mathbf{Y}_i \sim \mathcal{N}_p({\boldsymbol \mu}, {\boldsymbol \Sigma}) \text{ with } i \in \{1,\ldots,n\}$.

When defining the n × p matrix $\mathbf{X}$ for which the ith row is $\mathbf{Y}_i$, we obtain:

$\mathbf{X} \sim \mathcal{MN}_{n \times p}(\mathbf{M}, \mathbf{U}, \mathbf{V})$

where each row of $\mathbf{M}$ is equal to ${\boldsymbol \mu}$, that is $\mathbf{M}=\mathbf{1}_n \times {\boldsymbol \mu}^T$, $\mathbf{U}$ is the n × n identity matrix, that is the rows are independent, and $\mathbf{V} = {\boldsymbol \Sigma}$.

Relation to other distributions

Dawid (1981) provides a discussion of the relation of the matrix-valued normal distribution to other distributions, including the Wishart distribution, Inverse Wishart distribution and matrix t-distribution, but uses different notation from that employed here.

See also

• Multivariate normal distribution.

References

• Dawid, A.P. (1981). "Some matrix-variate distribution theory: Notational considerations and a Bayesian application". Biometrika 68 (1): 265–274. doi:10.1093/biomet/68.1.265. JSTOR 2335827. MR 614963. 
• Dutilleul, P (1999). "The MLE algorithm for the matrix normal distribution". Journal of Statistical Computation and Simulation 64 (2): 105–123. doi:10.1080/00949659908811970. 
• Arnold, S.F. (1981), The theory of linear models and multivariate analysis, New York: John Wiley & Sons, ISBN 0471050652