I am trying to calculate the Covariance matrix of the following random vector:
$Y = X-AX$
where $X$ is a random vector with $n$ entries and $A: n\times n$ transformational matrix where each entry is in the interval $[0,1]$.
I also know that the mean of all entries in $X$ is zero: $E(X) = 0$ and I know the Covariance matrix of $X$ as $\Sigma_X$.
I feel like I have everything to calculate $\Sigma_Y$ but I can't really wrap my head around it.
Thanks in advance :-)
Applying covariance definition is enough. Let's start computing first moments of $Y$: $$E[Y] = E[X-AX] = (I-A)E[X] = 0$$
\begin{align*} E[YY^T] & = E[(X-AX)(X-AX)^T] \\ & = E[XX^T-XX^TA^T-AXX^T+AXX^TA^T] \\ & = \Sigma_X-\Sigma_XA^T-A\Sigma_X+A\Sigma_XA^T \\ \end{align*} Now, using these expressions you can handle $\Sigma_Y$ \begin{align*} \Sigma_Y&=E[YY^T] = E[(Y-E[Y])(Y-E[Y])^T]\\ & = E[YY^T]-E[Y]E[Y]^T\\ & = \Sigma_X-\Sigma_XA^T-A\Sigma_X+A\Sigma_XA^T\\ \end{align*}