Derivative of $\log(\det(X+X^T)/2 )$ with respect to $X$

Question

I learned that the derivative of $\log(\det X)$ with respect to $X$ is $X^{-1}$. However, the following

Calculate $$\dfrac{d(\log(\det(X+X^T)/2 ))}{dX}$$

makes me confused. Could somebody help me? Thank you!

Answer 1

You need to use (called Jacobi formula) that $$ \frac{\partial \det(X)}{\partial x_{i,j}} = adj(X)^T_{i,j}, \text{ or equivalently } \ \frac{d \det(X)}{d X} = adj(X)^T. $$ the above comes from the following identity $$ A \cdot adj(A)^T = \det(A)\quad \forall\ A\in M(\mathbb{R},n).$$ So that you can compute $$ \frac{d \log(\det(X+X^T)/2)}{d X} = \frac{adj(X+X^T)^T}{\det(X+X^T)} = (X+X^T)^{-1}. $$