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Edit: Regarding the Wiki definition (the second one), you can write the random matrix as $$ X = U^{1/2}Z [V^{1/2}]^\intercal + M \tag{1}, $$ where $Z$ is a matrix of standard normals. When you take the $\text{vec}$ on both sides of that, you get $$ \text{vec}(X) - \text{vec}(M) = \text{vec}(X-M) = (V^{1/2} \otimes U^{1/2}) \text{vec}(Z) . $$ This is a ...


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