I encounter a problem as below: $$\mathcal{L}(A,\Sigma_{\xi})=\text{Tr}[(A-I)\Sigma_X(A-I)^\top+\Sigma_{\xi}].$$
One approach I'm trying is to replace $(A, \Sigma_{\xi})$ with $(\tilde A, I)$ with $\tilde A=\sqrt{D^{-1}}VA$ and $\Sigma_{\xi}=VDV^\top$ so that the number of variables to be optimized could be reduced. I want to see whether such a replacement would change the value of the objective function:
\begin{align} &\text{Tr}[(\tilde A-I)\Sigma_X(\tilde A -I)+I]\\ =&\text{Tr}[\sqrt{D^{-1}}VA\Sigma_XA^\top V^\top\sqrt{D^{-T}}-\sqrt{D^{-1}}VA\Sigma_X-\Sigma_XA^\top V^\top\sqrt{D^{-T}}+\Sigma_X+I]\\ =&\text{Tr}[\sqrt{D^{-1}}V(A\Sigma_XA^\top-A\Sigma_X\sqrt{D^{T}}V-V^\top\sqrt{D}\Sigma_XA^\top +V^\top\sqrt{D}\Sigma_X\sqrt{D^{T}}V+\Sigma_{\xi})V^\top\sqrt{D^{-T}}]\\ =&\text{Tr}[\sqrt{D^{-1}}V((A-V^\top\sqrt{D})\Sigma_X(A-V^\top\sqrt{D})^\top+\Sigma_{\xi})V^\top\sqrt{D^{-T}}]. \end{align} Is there any way to further simplify the equations? Thanks!