TITLE: Derivative of log-determinant of square root of a matrix
QUESTION [0 upvotes]: Let $\Sigma$ be a symmetric positive-definite matrix. We then denote $\Sigma^{1/2}$ as its square root so that $(\Sigma^{1/2})^2=\Sigma$. I am wondering if 
$$
\frac{\partial\ln[\det(\Sigma^{1/2}A\Sigma^{1/2})]}{\partial\Sigma}
$$
has a closed-form solution. Here, $A$ is a diagonal matrix with appropriate dimension.

REPLY [0 votes]: First note that$$\det(\Sigma^{1/2}A\Sigma^{1/2})=\det(A\Sigma^{1/2}\Sigma^{1/2})=\det(A\Sigma)=\det A\det\Sigma.$$Jacobi's formula implies $\partial_\Sigma\det\Sigma=(\operatorname{adj}\Sigma)^T$, so $$\partial_\Sigma\ln\det(\Sigma^{1/2}A\Sigma^{1/2})=\partial_\Sigma\ln\det\Sigma=(\det\Sigma)^{-1}(\operatorname{adj}\Sigma)^T.$$
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