I have the following function
$$A \mapsto -N\text{ln}\pi+N_{d}\text{ln}(\text{det}(A))-\sum_{n=1}^{N_d}\textbf{y}_n^{H}A\textbf{y}_n.$$
and I want to find its second order derivative with respect to $A$. I found that the second order derivative of first and third term will be zero. But what will come for second term?
Any help in this regard will be highly appreciated.
It's a well know result (and not difficult to prove) that
$$\frac{\partial}{\partial a_{ij}} |A| = (\operatorname{adj} A)_{ji}$$
or
$$\frac{\partial |A|}{\partial A} = (\operatorname{adj} A)^t$$
Hence, by the chain rule and Cramer's rule:
$$\frac{\partial \log |A|}{\partial A} = \frac{1}{|A|}(\operatorname{adj} A)^t= ({A^{-1}})^t$$