Suppose $X$ is multivariate Gaussian random variable $X\sim \mathcal{N}\left(0,H\right)$ and we define a new random variable $\eta$ by it's multiplication with some another random variable $Y$: $\eta = YX$ What should I need to require from $Y$ in order that $\eta$ will be Gaussian but also uncorrelated with $X$?

Suppose $X$ is a multivariate Gaussian random variable $X\sim \mathcal{N}\left(0,H\right)$ and we define a new random variable $\eta$ by its multiplication with some other random variable $Y$, i.e., $\eta = YX$.

What should I need to require from $Y$ in order that $\eta$ will be Gaussian but will also be uncorrelated with $X$?

Suppose $X$ is multivariate Gaussian random variable $X\sim \mathcal{N}\left(0,H\right)$ and we define a new random variable $\eta$ by it's multiplication with some another random variable $Y$: $\eta = YX$ What should I need to require from $Y$ in order that $\eta$ will be Gaussian but also uncorrected with $X$?

Suppose $X$ is multivariate Gaussian random variable $X\sim \mathcal{N}\left(0,H\right)$ and we define a new random variable $\eta$ by it's multiplication with some another random variable $Y$: $\eta = YX$ What should I need to require from $Y$ in order that $\eta$ will be Gaussian but also uncorrelated with $X$?

suppose $X$ is multivariate Gaussian random variable $X\sim \mathcal{N}\left(0,H\right)$ and we define a new random variable $\eta$ by it's multiplication with some another random variable $Y$: $\eta = YX$ What should I need to require from $Y$ in order that $\eta$ will be gaussian but also uncorrected with $X$?

Suppose $X$ is multivariate Gaussian random variable $X\sim \mathcal{N}\left(0,H\right)$ and we define a new random variable $\eta$ by it's multiplication with some another random variable $Y$: $\eta = YX$ What should I need to require from $Y$ in order that $\eta$ will be Gaussian but also uncorrected with $X$?