The Matrix-Gaussian distribution



Gupta and Nagar (1999):

The random matrix \(X(p \times n)\) is said to have a matrix variate normal distribution with mean matrix \(M(p \times n)\) and covariance matrix \(\Sigma \otimes \Psi\) where \(\Sigma(p \times p)>0\) and \(\Psi(n \times n)>0\), if \(\operatorname{vec}\left(X^{\prime}\right) \sim N_{p n}\left(\operatorname{vec}\left(M^{\prime}\right), \Sigma \otimes \Psi\right)\)

We shall use the notation \(X \sim N_{p, n}(M, \Sigma \otimes \Psi)\).

They prove the following theorem:

If \(X \sim N_{p, n}(M, \Sigma \otimes \Psi)\), then the p.d.f. of \(X\) is given by \[ \begin{array}{r} (2 \pi)^{-\frac{1}{2} n p} \operatorname{det}(\Sigma)^{-\frac{1}{2} n} \operatorname{det}(\Psi)^{-\frac{1}{2} p} \operatorname{etr}\left\{-\frac{1}{2} \Sigma^{-1}(X-M) \Psi^{-1}(X-M)^{\prime}\right\} \\ X \in \mathbb{R}^{p \times n}, M \in \mathbb{R}^{p \times n} \end{array} \]

References

Gupta, A. K., and D. K. Nagar. 1999. Matrix Variate Distributions. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 104. Boca Raton: Chapman and Hall/CRC.
———. 2018. Matrix Variate Distributions. CRC Press.

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