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Wikipedia has the following definition of a normal random vector:

A real random vector $\mathbf{X} = (X_1,\ldots,X_k)^{\mathrm T}$ is called a normal random vector if there exists a random $\ell$-vector $\mathbf{Z}$, which is a standard normal random vector, a $k$-vector $\mathbf{\mu}$, and a $k \times \ell$ matrix $\boldsymbol{A}$, such that $\mathbf{X}=\boldsymbol{A} \mathbf{Z} + \mathbf{\mu}$. Then one writes $X \sim \mathcal N_k(\mathbf \mu, \mathbf \Sigma)$ with $\mathbf \Sigma = \mathbf A \mathbf A^T$.

Does an equivalent representation exist for a normal random matrix, i.e. a normal random matrix as a linear transformation of a standard normal random matrix?

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  • $\begingroup$ There's nothing to show or prove, because a Normal random matrix simply is a Normal random vector arranged in a tabular form: the distinction is merely a matter of notation. That's what the Wikipedia article is trying to tell you at the line "The matrix normal is related to the multivariate normal distribution..." $\endgroup$
    – whuber
    Apr 4 '20 at 22:02
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A normal random matrix $\mathbf{X} \sim \mathcal{MN}_{n\times p}(\mathbf{M}, \mathbf{U}, \mathbf{V})$ is transformed as

$$\mathbf{DXC}\sim \mathcal{MN}_{r\times s}(\mathbf{DMC}, \mathbf{DUD}^T, \mathbf{C}^T\mathbf{VC})$$

for $\mathbf D, \mathbf C$ full rank.

So with the "standard normal random matrix" $\mathbf Z \sim \mathcal N_{n\times p}(0, I_n, I_p)$ one can write

$$\mathbf X = \mathbf M + \mathbf A \mathbf Z \mathbf B \sim \mathcal N_{n\times p}(\mathbf M + \mathbf A\mathbf 0\mathbf B, \mathbf A\mathbf I_n\mathbf A^T, \mathbf B \mathbf I_p\mathbf B^T) = \mathcal N_{n \times p}(\mathbf M, \underbrace{\mathbf U}_{=\mathbf A\mathbf A^T}, \underbrace{\mathbf V}_{=\mathbf B\mathbf B^T}).$$

On second read, the above was more or less stated on Wikipedia.

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