# 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 $\begin{array}{r} (2 \pi)^{-\frac{1}{2} n p} \operatorname{det}(\Sigma)^{-\frac{1}{2} n} \operatorname{det}(\Psi)^{-\frac{1}{2} p} \operatorname{etr}\left\{-\frac{1}{2} \Sigma^{-1}(X-M) \Psi^{-1}(X-M)^{\prime}\right\} \\ X \in \mathbb{R}^{p \times n}, M \in \mathbb{R}^{p \times n} \end{array}$

Is this the same matrix normal as discussed in scipy.stats.matrix_normal? If so

The probability density function for matrix_normal is $f(X)=(2 \pi)^{-\frac{m n}{2}}|U|^{-\frac{n}{2}}|V|^{-\frac{m}{2}} \exp \left(-\frac{1}{2} \operatorname{Tr}\left[U^{-1}(X-M) V^{-1}(X-M)^T\right]\right)$ where $$M$$ is the mean, $$U$$ the among-row covariance matrix, $$V$$ the among-column covariance matrix. The allow_singular behaviour of the multivariate_normal distribution is not currently supported. Covariance matrices must be full rank. The matrix_normal distribution is closely related to the multivariate_normal distribution. Specifically, $$\operatorname{Vec}(X)$$ (the vector formed by concatenating the columns of $$X$$ ) has a multivariate normal distribution with mean $$\operatorname{Vec}(M)$$ and covariance $$V \otimes U$$ (where $$\otimes$$ is the Kronecker product). Sampling and pdf evaluation are $$\mathcal{O}\left(m^3+n^3+m^2 n+m n^2\right)$$ for the matrix normal, but $$\mathcal{O}\left(m^3 n^3\right)$$ for the equivalent multivariate normal, making this equivalent form algorithmically inefficient.

## 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.

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