Compute the derivative of the log of the determinant of A with respect to A

Question

Let $A$ be a positive definite matrix. Compute the derivative of $\log \det A$ with respect to $A$.

Here's what I have thus far.

\begin{align*} \frac{d}{dA}\log (\det (A)) &= \frac{1}{det(A)} \frac{d}{dA}(det(A))~ \text{, By chain rule}\\ \end{align*}

I know that $\frac{1}{det(A)} \frac{d}{dA_{i,j}}(det(A)) = \frac{1}{det(A)} (adj(A)_{i,j})^T$. Additionally that $A^{-1} = \frac{1}{det(A)} adj(A)$.

I'm under the impression that the solution is $\frac{d}{dA}\log (\det (A)) = 2A^{-1} - (A^{-1} \circ I)$. Where $\circ$ is Hadamard product.

I'm missing some understanding about the adjoint matrix to connect it to the solution.

Answer 1

The derivative of the map $f:A\mapsto\log\det A$ is the linear map$$df_A(H)=\left.\frac{d}{dt}\right|_{t=0}\log\det (A+tH)=\frac{1}{\det A}\cdot\det A\cdot \operatorname{trace}(A^{-1}H)=\operatorname{trace}(A^{-1}H),$$ where I used the chain rule and a formula for the Derivative of a determinant.