assume two dependent random variables X and Y with their covariance matrix $\Sigma_{ij}$. Now assume one multiplies X with another independent random variable Z. The variance of the resulting random variable of the product is then given by (correct me if I am wrong):
$Var[X*Z] = (E[X])^2Var[Z] + (E[Z])^2Var[X] + Var[X]Var[Z]$
My questions is: How does this multiplication influence the covariance matrix $\Sigma_{ij}$? Intuitively I would say it just influences the element $\Sigma_{XX}$ on the main diagonal of $\Sigma$ (replaced by $Var[X*Z]$). But I am not quite sure how to show that.
Thanks for yours help :-)
Best