Matrix Normal Distribution - Definition

Definition

The probability density function for the random matrix X (n × p) that follows the matrix normal distribution has the form:

\frac{\exp\left( -\frac{1}{2} \mbox{tr}\left \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\;,\;
{\boldsymbol \Omega} = E/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:

if and only if

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

where denotes the Kronecker product and denotes the vectorization of .

Read more about this topic:  Matrix Normal Distribution

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