We have that: $\textbf Y \in \mathbb{R}^{n \times q}, \textbf G \in \mathbb{R}^{n \times n}, \textbf P \in \mathbb{R}^{n \times n}, \textbf Q \in \mathbb{R}^{q \times q}$. Furthermore, $\textbf G$ is a diagonal matrix, $\textbf P$ is orthogonal and $\textbf Q$ is positive definite.
Given: \begin{equation} |\textbf Y^\top \textbf P \textbf G \textbf P^\top \textbf Y + \textbf Q|^{-\frac{q}{2}} \tag{1} \end{equation} I am interested in splitting (1) into a product of 2 functions such that $\textbf G$ is separate from $\textbf Y$. I don't care where the other matrices end up, just as long as $\textbf G$ and $\textbf Y$ appear in separate products. Is this possible?
(1) is also actually proportional to a function of $\textbf Y$ only, so up to proportionality, I can take out any factors of the determinant which don't include $\textbf Y$. For example, instead of (1), I can work with: \begin{equation} (1) \propto |\textbf Q^{-1} \textbf Y^\top \textbf P \textbf G \textbf P^\top \textbf Y + \textbf I_{q}|^{-\frac{q}{2}} \end{equation}
I've tried loads of different things, but I can't seem to split up $\textbf G$. I have a feeling that this won't be possible but I was wondering whether there exist any nice theorems to help me do this? Thank you!