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 $\mathrm{\Sigma }\otimes \mathrm{\Psi }$ where $\mathrm{\Sigma }(p\times p)>0$ and $\mathrm{\Psi }(n\times n)>0$, if $\mathrm{vec}\left(X^{\prime }\right)\sim N_{pn}(\mathrm{vec}\left(M^{\prime }\right),\mathrm{\Sigma }\otimes \mathrm{\Psi })$

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

They prove the following theorem:

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

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

The probability density function for matrix_normal is $$f(X)=(2\pi {)}^{-\frac{mn}{2}}|U{|}^{-\frac{n}{2}}|V{|}^{-\frac{m}{2}}\exp (-\frac{1}{2}\mathrm{Tr}[U^{-1}(X-M)V^{-1}(X-M{)}^T])$$ 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, $\mathrm{Vec}(X)$ (the vector formed by concatenating the columns of $X$) has a multivariate normal distribution with mean $\mathrm{Vec}(M)$ and covariance $V\otimes U$ (where $\otimes$ is the Kronecker product). Sampling and pdf evaluation are $\mathcal{O}(m^3+n^3+m^{2}n+m{n}^2)$ for the matrix normal, but $\mathcal{O}\left(m^3{n}^3\right)$ for the equivalent multivariate normal, making this equivalent form algorithmically inefficient.

Looks right. For an actual introduction, the section in The Book of Statistical Proof proves some useful theorems in a consistent notation.