# Is Gaussian $(X_1, X_2)$ optimal for $h(a_1X_1+ a_2X_2+Z_1) - \mu \, h( b_1X_1+b_2X_2+ Z_2)$?

Let \begin{align} W &= h(X_1+Z_1) - \mu \, h( X_2+ Z_2) \quad (1) \end{align} where $h(\cdot)$ is the differential entropy function, $\mu\ge 1$ is a scalar, and $Z_1$ and $Z_2$ are independent and have Gaussian distributions. If $X=[X_1 \; X_2]^t$ is independent of $Z_1$ and $Z_2$, we know that a Gaussian $X$ is an optimal solution for \begin{aligned} %{\mathcal W_0} =\; &\underset{p(x)}{\text{maximize}} & & W\\ & \text{subject to} & & \operatorname{Cov}(X) \preceq S \end{aligned} in which $S$ is a positive semi-definite matrix.

My question is whether from the above we can deduce that \begin{aligned} %{\mathcal W_0} =\; &\underset{p(x)}{\text{maximize}} & & W' = h(a_1X_1+ a_2X_2+Z_1) - \mu h( b_1X_1+b_2X_2+ Z_2) \\ & \text{subject to} & & \operatorname{Cov}(X)\preceq S \end{aligned} where $a_1, a_2, b_1, b_2$ are scalars, is also maximized by a Gaussian $X$? In other words, in (1), can we simply replace $X_1$ and $X_2$ by $a_1X_1+a_2X_2$ and $b_1X_1+b_2X_2$? I assume $a_1b_2-a_2b_1\neq 0.$