I am considering a two-dimensional linear SDE of the form $$\begin{cases} dX_1 =\bigl(A_{11} X_1 + A_{12} X_2 \bigr)\,dt + \sigma \,dW_1, \\ dX_2 =\bigl(A_{21} X_1 + A_{22} X_2 \bigr)\,dt + \sigma \,dW_2, \end{cases} $$ where $W_1$ and $W_2$ are one-dimensional Wiener processes, the $A_{ij}$ are real, and so is $\sigma$.
My goal is to get an expression for the PSD of $X_1$. I have looked at both suggestions in this post but I am unable to apply them to my problem. Having two Wiener processes $W_1$ and $W_2$ (instead of one as in the referenced post) throws me off. Any help would be greatly appreciated, thank you.