Suppose we have the matrices $Z\in \mathbb{R}^{n\times n}$ and $X\in \mathbb{R}^{n\times d}$, such that each row $x_i\in\mathbb{R}^d$ is drawn i.i.d from a $N(0,\Sigma_{d\times d})$ distribution. What is the distribution of $ZX$?

Would it be,

\begin{equation} ZX\stackrel{?}{\sim} N\left(0,Z_{n\times n}\otimes \Sigma_{d\times d}\right) \end{equation}

  • 1
    $\begingroup$ As a quick reality check, contemplate your solution when $n=d=1.$ $\endgroup$
    – whuber
    Nov 24 at 18:20

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