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{{Probability distribution
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| name      =Matrix normal
| type      =density
| box_width  =195px
| pdf_image  =
| cdf_image  =
| notation  =<math>\mathcal{MN}_{n,p}(\mathbf{M}, \mathbf{U}, \mathbf{V})</math>
| parameters =<math>\mathbf{M}</math> [[location parameter|location]] ([[real number|real]] <math>n\times p</math> [[matrix (mathematics)|matrix]])<br/>
<math>\mathbf{U}</math> [[scale matrix|scale]] ([[positive-definite matrix|positive-definite]] [[real number|real]] <math>n\times n</math> [[matrix (mathematics)|matrix]])<br/>
<math>\mathbf{V}</math> [[scale matrix|scale]] ([[positive-definite matrix|positive-definite]] [[real number|real]] <math>p\times p</math> [[matrix (mathematics)|matrix]])
| support    =<math>\mathbf{X} \in \mathbb{R}^{n \times p}</math>
| pdf        =<math>\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}}</math>
| cdf        =
| mean      =<math>\mathbf{M}</math>
| median    =
| mode      =
| variance  =<math>\mathbf{U}</math> (among-row) and <math>\mathbf{V}</math> (among-column)
| skewness  =
| kurtosis  =
| entropy    =
| mgf        =
| char      =
}}


In [[statistics]], the '''matrix normal distribution''' is a [[probability distribution]] that is a generalization of the [[multivariate normal distribution]] to matrix-valued random variables.
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== Definition ==
The [[probability density function]] for the random matrix '''X''' (''n''&nbsp;&times;&nbsp;''p'') that  follows the matrix normal distribution <math>\mathcal{MN}_{n,p}(\mathbf{M}, \mathbf{U}, \mathbf{V})</math> has the form:
 
:<math>
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}}
</math>
 
where '''M''' is ''n''&nbsp;&times;&nbsp;''p'', '''U''' is ''n''&nbsp;&times;&nbsp;''n'' and '''V''' is ''p''&nbsp;&times;&nbsp;''p''.
 
There are several ways to define the two covariance matrices. One possibility is
 
:<math>\mathbf{U} = E[(\mathbf{X} - \mathbf{M})(\mathbf{X} - \mathbf{M})^{T}]</math>
 
:<math>\mathbf{V} = E[(\mathbf{X} - \mathbf{M})^{T} (\mathbf{X} - \mathbf{M})] / c</math>
 
where <math>c</math> 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:
 
:<math>\mathbf{X} \sim \mathcal{MN}_{n\times p}(\mathbf{M}, \mathbf{U}, \mathbf{V}),</math>
 
if and only if
 
:<math>\mathrm{vec}(\mathbf{X}) \sim \mathcal{N}_{np}(\mathrm{vec}(\mathbf{M}), \mathbf{V} \otimes \mathbf{U})</math>
 
where <math>\otimes</math> denotes the [[Kronecker product]] and <math>\mathrm{vec}(\mathbf{M})</math> denotes the [[vectorization (mathematics)|vectorization]] of <math>\mathbf{M}</math>.
 
==Example==
Let's imagine a sample of ''n'' independent ''p''-dimensional random variables identically distributed according to a [[multivariate normal distribution]]:
:<math>\mathbf{Y}_i \sim \mathcal{N}_p({\boldsymbol \mu}, {\boldsymbol \Sigma}) \text{ with } i \in \{1,\ldots,n\}</math>.
When defining the ''n''&nbsp;&times;&nbsp;''p'' matrix <math>\mathbf{X}</math> for which the ''i''th row is <math>\mathbf{Y}_i</math>, we obtain:
:<math>\mathbf{X} \sim \mathcal{MN}_{n \times p}(\mathbf{M}, \mathbf{U}, \mathbf{V})</math>
where each row of <math>\mathbf{M}</math> is equal to <math>{\boldsymbol \mu}</math>, that is <math>\mathbf{M}=\mathbf{1}_n \times {\boldsymbol \mu}^T</math>, <math>\mathbf{U}</math> is the ''n''&nbsp;&times;&nbsp;''n'' identity matrix, that is the rows are independent, and <math>\mathbf{V} = {\boldsymbol \Sigma}</math>.
 
==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==
* {{cite journal
|last=Dawid |first=A.P. |authorlink=Philip Dawid
|year=1981
|title=Some matrix-variate distribution theory: Notational considerations and a Bayesian application
|journal=[[Biometrika]]
|volume=68 |issue=1 |pages=265&ndash;274
|doi=10.1093/biomet/68.1.265  |mr=614963 | jstor = 2335827
}}
* {{cite journal
|last=Dutilleul |first=P
|year=1999
|title=The MLE algorithm for the matrix normal distribution
|journal=[[Journal of Statistical Computation and Simulation]]
|volume=64 |issue=2 |pages=105&ndash;123
|doi=10.1080/00949659908811970
}}
* {{Citation
|last=Arnold |first=S.F.
|title=The theory of linear models and multivariate analysis
|publisher=[[John Wiley & Sons]]
|place=New York
|year=1981
|isbn=0471050652
}}
 
{{ProbDistributions|multivariate}}
 
[[Category:Random matrices]]
[[Category:Continuous distributions]]
[[Category:Multivariate continuous distributions]]
[[Category:Probability distributions]]

Latest revision as of 19:06, 10 January 2015

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