I have looked at this question already. However, my question is different.
Suppose $X\sim N(A\mu, C\Sigma C^T)$ (multivariate normal), where $\mu$ is $n \times 1 $, $A$ is $n \times n$, $C$ is $n \times p$ and $\Sigma$ is $p \times p$.
The log-loglikelihood that I derived is as follow (ignoring constants):
$$\ell = -\log \Big( \det \big[C \Sigma C^T \big] \Big) - (x-A\mu)^T (C \Sigma C^T)^{-1}(x-A\mu)$$.
My Question is: how can I differentiate the log-likelihood with respect to not only $\Sigma$ but also $A$ and $C$ ? i.e. How can I get the following
- $$\frac{\partial \ell}{\partial\Sigma}$$
- $$\frac{\partial \ell}{\partial A}$$
- $$\frac{\partial \ell}{\partial C}$$
And how should the terms be rearranged such that $A, C, \Sigma$ will be on one side after differentiation ?
