# Normal random matrix as linear transformation of standard normal random matrix?

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?

• 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..."
– whuber
Apr 4 '20 at 22:02

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