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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.

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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 $S^{-1}$.

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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.

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  • $\begingroup$ Ohh. Thanks. That was obvious. Sorry! $\endgroup$ Sep 6 '16 at 15:51

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