Matrix normal distribution

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The matrix normal distribution is a probability distribution that is a generalization of the normal distribution. 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}) =(2\pi)^{-np/2} |{\boldsymbol \Omega}|^{-n/2} |{\boldsymbol \Sigma}|^{-p/2} \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).

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}.

[edit] See also

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