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| {{Probability distribution
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| | name =Matrix normal
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| | type =density
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| | box_width =195px
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| | pdf_image =
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| | cdf_image =
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| | notation =<math>\mathcal{MN}_{n,p}(\mathbf{M}, \mathbf{U}, \mathbf{V})</math>
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| | parameters =<math>\mathbf{M}</math> [[location parameter|location]] ([[real number|real]] <math>n\times p</math> [[matrix (mathematics)|matrix]])<br/>
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| <math>\mathbf{U}</math> [[scale matrix|scale]] ([[positive-definite matrix|positive-definite]] [[real number|real]] <math>n\times n</math> [[matrix (mathematics)|matrix]])<br/>
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| <math>\mathbf{V}</math> [[scale matrix|scale]] ([[positive-definite matrix|positive-definite]] [[real number|real]] <math>p\times p</math> [[matrix (mathematics)|matrix]])
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| | support =<math>\mathbf{X} \in \mathbb{R}^{n \times p}</math>
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| | 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>
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| | cdf =
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| | mean =<math>\mathbf{M}</math>
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| | median =
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| | mode =
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| | variance =<math>\mathbf{U}</math> (among-row) and <math>\mathbf{V}</math> (among-column)
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| | skewness =
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| | kurtosis =
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| | entropy =
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| | mgf =
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| | char =
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| }}
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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.
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| == Definition ==
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| The [[probability density function]] for the random matrix '''X''' (''n'' × ''p'') that follows the matrix normal distribution <math>\mathcal{MN}_{n,p}(\mathbf{M}, \mathbf{U}, \mathbf{V})</math> has the form:
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| :<math>
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| 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}}
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| </math>
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| where '''M''' is ''n'' × ''p'', '''U''' is ''n'' × ''n'' and '''V''' is ''p'' × ''p''.
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| There are several ways to define the two covariance matrices. One possibility is
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| :<math>\mathbf{U} = E[(\mathbf{X} - \mathbf{M})(\mathbf{X} - \mathbf{M})^{T}]</math> | |
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| :<math>\mathbf{V} = E[(\mathbf{X} - \mathbf{M})^{T} (\mathbf{X} - \mathbf{M})] / c</math>
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| where <math>c</math> is a constant which depends on '''U''' and ensures appropriate power normalization.
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| The matrix normal is related to the [[multivariate normal distribution]] in the following way:
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| :<math>\mathbf{X} \sim \mathcal{MN}_{n\times p}(\mathbf{M}, \mathbf{U}, \mathbf{V}),</math>
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| if and only if
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| :<math>\mathrm{vec}(\mathbf{X}) \sim \mathcal{N}_{np}(\mathrm{vec}(\mathbf{M}), \mathbf{V} \otimes \mathbf{U})</math>
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| 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>.
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| ==Example==
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| Let's imagine a sample of ''n'' independent ''p''-dimensional random variables identically distributed according to a [[multivariate normal distribution]]:
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| :<math>\mathbf{Y}_i \sim \mathcal{N}_p({\boldsymbol \mu}, {\boldsymbol \Sigma}) \text{ with } i \in \{1,\ldots,n\}</math>.
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| When defining the ''n'' × ''p'' matrix <math>\mathbf{X}</math> for which the ''i''th row is <math>\mathbf{Y}_i</math>, we obtain:
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| :<math>\mathbf{X} \sim \mathcal{MN}_{n \times p}(\mathbf{M}, \mathbf{U}, \mathbf{V})</math>
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| 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'' × ''n'' identity matrix, that is the rows are independent, and <math>\mathbf{V} = {\boldsymbol \Sigma}</math>.
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| ==Relation to other distributions==
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| 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.
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| == See also ==
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| * [[Multivariate normal distribution]].
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| ==References==
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| * {{cite journal
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| |last=Dawid |first=A.P. |authorlink=Philip Dawid
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| |year=1981
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| |title=Some matrix-variate distribution theory: Notational considerations and a Bayesian application
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| |journal=[[Biometrika]]
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| |volume=68 |issue=1 |pages=265–274
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| |doi=10.1093/biomet/68.1.265 |mr=614963 | jstor = 2335827
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| }}
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| * {{cite journal
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| |last=Dutilleul |first=P
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| |year=1999
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| |title=The MLE algorithm for the matrix normal distribution
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| |journal=[[Journal of Statistical Computation and Simulation]]
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| |volume=64 |issue=2 |pages=105–123
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| |doi=10.1080/00949659908811970
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| }}
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| * {{Citation
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| |last=Arnold |first=S.F.
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| |title=The theory of linear models and multivariate analysis
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| |publisher=[[John Wiley & Sons]]
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| |place=New York
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| |year=1981
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| |isbn=0471050652
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| }}
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| {{ProbDistributions|multivariate}}
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| [[Category:Random matrices]]
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| [[Category:Continuous distributions]]
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| [[Category:Multivariate continuous distributions]]
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| [[Category:Probability distributions]]
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