What is $\frac{\partial}{\partial T}\left[\mathrm{ln}\left(\mathrm{det}\left( TQT^T+BI\right ) \right ) \right ]$ (and how do I derive the answer in an intuitive way)? $T\in\mathbb{R}^{m\times n},Q_l\in\mathbb{R}^{n\times n},B\in\mathbb{R}$, and $I$ is the $n\times n$ identity matrix, so the answer should be $\in\mathbb{R}^{m\times n}$.
For context, this arises from the following problem: $y=Tx+v$, with $y\in\mathbb{R}^{m},x\in\mathbb{R}^{n},v\in\mathbb{R}^m$, and $x$ and $v$ are random variables with distributions $p(x)=\mathcal{N}\left(x|\mu_x,Q \right )$ and $p(v)=\mathcal{N}\left(v|0,BI \right )$ respectively; how do I choose $T$ such that the mutual information between $x$ and $y$ is maximised (subject to some arbitrary constraint or regularisation term that is not included here, for example all the rows of $T$ have fixed Euclidean norm)? The mutual information can be expressed as $\mathbb{E}\left[\mathrm{ln}\left(\frac{p(y|x)}{p(y)} \right )\right]$, which in this case involving Gaussian random variables simplifies to $\frac{1}{2}\mathrm{ln}\left(\frac{\det\left(\mathrm{Var}(y) \right )}{\det\left(\mathrm{Var}(y|x) \right )} \right ) = \frac{1}{2}\mathrm{ln}\left(\frac{\det\left(TQT^T+BI \right )}{\det\left(BI \right )} \right )$, so I'm looking for the gradient in question in order to proceed with this optimisation. Thanks in advance for your help.
$ \def\l{\lambda}\def\s{\sigma}\def\e{\varepsilon} \def\LR#1{\left(#1\right)} \def\trace#1{\operatorname{Tr}\LR{#1}} \def\sym#1{\operatorname{sym}\LR{#1}} \def\qiq{\quad\implies\quad} \def\p{\partial} \def\grad#1#2{\frac{\p #1}{\p #2}} \def\c#1{\color{blue}{#1}} $For typing convenience, define the matrix variable $$\eqalign{ F &= {TQT^T + BI} \\ }$$ Rewrite the function, calculate its differential using Jacobi's formula, then extract the gradient $$\eqalign{ \l &= \log(\det(F)) \\ \c{d\l} &\c{=} \c{\trace{F^{-1}dF}} \\ &= \trace{F^{-1}\LR{dT\:QT^T+TQ\:dT^T}} \\ &= \trace{2\,QT^TF^{-1}\,dT} \\ \grad{\l}{T} &= 2\,QT^TF^{-1} \\ }$$ Depending on your preferred layout convention, you may wish to transpose this result.