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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 ${\mathcal{ℳ}\mathcal{𝒩}}_{n,p}(\mathbf{M},\mathbf{U},\mathbf{V})$ has the form:

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

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:

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

if and only if

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

where $\otimes$ denotes the Kronecker product and $\mathrm{v}\mathrm{e}\mathrm{c}(\mathbf{M})$ 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:

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

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

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

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

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

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