Derivative of log determinant [closed]

Question

Let $x_i \in\mathbb{R}^d$ and $a_i\in [0,1]$ for $i = 1,\dots,k$. How to compute the following derivative?

$$ \frac{d}{da_j}\log \det\left(\sum_{i = 1}^k a_ix_ix_i^\top\right). $$

Answer 1

\begin{align*} &\frac{d}{da_j}\log \det\left(\sum_{i = 1}^k a_ix_ix_i^\top\right) \\ &= \frac1{\det\left(\sum_{i = 1}^k a_ix_ix_i^\top\right)}\det\left(\sum_{i = 1}^k a_ix_ix_i^\top\right)\operatorname{Tr}\left(\left(\sum_{i = 1}^k a_ix_ix_i^\top\right)^{-1}x_jx_j^\top\right) \\&= \operatorname{Tr}\left(\left(\sum_{i = 1}^k a_ix_ix_i^\top\right)^{-1}x_jx_j^\top\right) \end{align*} using $d \det(A)(X) = \operatorname{Tr}(\operatorname{adj}(A)X) = \det(A)\operatorname{Tr}(A^{-1}X)$.