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This question is motivated by this post. Then the log multivariate normal density looks like this:

\begin{equation} \implies \log(1)+ \frac{p}{2}\log(2\pi)+ \frac{1}{2}\log(|\Sigma^X_j+\Sigma^Y_{k}|^{-1})- \frac{1}{2}(Z_i-\mu^X_j)^T(\Sigma^X_j+\Sigma^Y_k)^{-1}(Z_i-\mu^X_j) \end{equation}

Now, One way is to use EM to update the estimates, but in M-step, one could encounter taking derivative of $\log(det(\Sigma^X_j+\Sigma^Y_{k})^{-1})$ or derivative of above equation respect to $\Sigma^X_j$ or $\Sigma^Y_k$, how can this be done? Many thanks in advance.

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