When reading about interior-point methods in Stephen Boyd & Lieven Vandenberghe's Convex Optimization, a question arose about how to use barrier method for the constraint $X$ is **positive definite**, i.e., $X \succ 0$.

Intuitively, we can add logarithmic barrier function ${\log\det(X)}$ to the objective function, but when calculating the Newton step, we have to get the derivative of ${\log\det(X)}$ and the second derivative of ${(\log \det(X))^{-1}}$. It is easy to find that the derivative ${\log\det(X)}$ is $X^{-1}$, but it seems quite difficult to get the second derivative. So, how to deal with such a problem?