What is the Hessian of log det X?

Question

I know that $f(X) = \log \det X$ is concave on domain $S^n_{++}$, but what is the Hessian of f(X)? Is there any book I can refer to?

Answer 1

I think people are confused because in your notation $$det(X)= |X|$$

Hint: To solve this we apply the chain rule. You have composed det with log.

The derivative of $\log$ is $d(\log(u))_y(h)=h/y$

The derivative $d(det(U))_{X}(H)= Tr(adj(X)XH)$

Applying chain rule gives:

$$ d(\log(det(U))_X(H)= \frac{1}{det(X)}Tr(adj(X)XH)$$

Now to complete you want to differentiate this expression with respect to X in the same way which I let you try, you will get a linear map in two matrices(it’s exactly the same idea), H and say K.

If you want a Hessian then you want to think about this as a map from $\mathbb{R}^{n^2}$ to $\mathbb{R}$ so plug in matrices H and K which form a natural basis (I.e. matrix with zero everywhere but position m,n) and read off the second order partials.

Edit:

To clarify, this second derivative is a set of $n^4$ numbers, I have suggested you arrange them as a hessian matrix by thinking of this as a map from $\mathbb{R}^{n^2}$ to $\mathbb{R}$. Note it is not a tensor as it won’t transform like one.