Matrix normal distribution

Matrix normal distribution

Parameters	$\mathbf{M}\,$ - mean $\mathbf{\Omega} \,$ - row covariance $\mathbf{\Sigma} \,$ - column covariance. Parameters are matrices (all of them).
Support	$\mathbf{W}\!$ is a matrix
PDF	(see Notebox below)
Mean	$\mathbf{M}$

Notebox	Probability density function: $\frac{\exp\left( -\frac{1}{2} \mbox{tr}\left[ {\boldsymbol \Omega}^{-1} (\mathbf{X} - \mathbf{M})^{T} {\boldsymbol \Sigma}^{-1} (\mathbf{X} - \mathbf{M}) \right] \right)}{(2\pi)^{np/2} |{\boldsymbol \Omega}|^{n/2} |{\boldsymbol \Sigma}|^{p/2}}$

The matrix normal distribution is a probability distribution that is a generalization of the 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 has the form:

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

where M is n × p, Ω is p × p and Σ is n × n. There are several ways to define the two covariance matrices. One possibility is

${\boldsymbol \Sigma} = E[ (\mathbf{X} - \mathbf{M})(\mathbf{X} - \mathbf{M})^{T}]\;,\;$

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

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

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

$\mathbf{X} \sim MN_{n\times p}(\mathbf{M}, {\boldsymbol \Omega}, {\boldsymbol \Sigma}),$

if and only if

$\mathrm{vec}\;\mathbf{X} \sim N_{np}(\mathrm{vec}\;\mathbf{M}, {\boldsymbol \Omega}\otimes{\boldsymbol \Sigma}),$

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

Example

Matrix Normal random variables arise from a sample identically distributed multivariate Normal random variables with possible dependence between the vectors. For example, if $\mathbf{Y}$ is an (n × p) matrix whose rows are independent with distribution $N_{p}(\mathbf{0}, {\boldsymbol \Sigma})$ then $\mathbf{Y} \sim MN_{n\times p}(\mathbf{0}, \mathbf{I}_n, {\boldsymbol \Sigma})$, where $\mathbf{I}_n$ is the (n × n) identity matrix. On the other hand, the columns of $\mathbf{X=}{\boldsymbol \Gamma}\mathbf{Y}$ are dependent but identically distributed multivariate Normal random variables. Furthermore, $\mathbf{X+M}\sim MN_{n\times p}(\mathbf{M}, {\boldsymbol \Omega, \Sigma})$, where ${\boldsymbol \Omega = \Gamma\Gamma}^T$.

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 a "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. MR614963.