# Distribution of a product of a matrix with a random matrix

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,

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

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