What is the derivative of $\mbox{trace} (\Sigma^.5 S^{-1}\Sigma^.5)$ with respect to $\Sigma$?

Question

While fitting a Gaussian distribution to another Gaussian, I came up with this term

$$\mbox{trace} (\Sigma^{\frac{1}{2}} S^{-1}\Sigma^{\frac{1}{2}})$$

I need to compute its derivative with respect to $\Sigma$. Both $\Sigma$ and $S$ are positive definite matrices.

Answer 1

The trace of product is invariant under cyclic permutations of the product. Thus

$\mbox{tr}(\Sigma^{1/2}S^{-1}\Sigma^{1/2})=\mbox{tr}(\Sigma^{1/2}\Sigma^{1/2}S^{-1})=\mbox{tr}(\Sigma S^{-1})$

The derivative of $\mbox{tr}(\Sigma S^{-1})$ with respect to $\Sigma$ is $S^{-1}$.

Answer 2

Note that $$ \operatorname{trace}(\Sigma^{1/2}S^{-1}\Sigma^{1/2}) = \operatorname{trace}(S^{-1}\Sigma) $$ It should be straightforward to compute this derivative, what ever your method.