I'm trying to apply Newton's method to find a local optimum of the matrix $\Sigma$ to minimize the objective function:
$f(\Sigma) = -\sum_{n=1}^{N}\left(-\ln{2\pi} - \frac{1}{2}\ln{|\Sigma|}-\frac{1}{2}\omega_n^T\Sigma\omega_n\right) - \left(\frac{\nu}{2}|\Phi| - \nu\ln{2} -\ln{\Gamma_2{(\frac{\nu}{2})}} -\frac{\nu+3}{2}\ln{|\Sigma|} - \frac{1}{2}tr{(\Phi\Sigma^{-1})}\right)$
Assume only $\Sigma$ is the variable in the equation. Newton's method approaches the local optimum by iterating this:
$x_{n+1} = x_n - [Hf(x_n)]^{-1}\nabla f(x_n)$
I'm confused on how to apply Newton's method to solve this problem. Because my variable is a matrix, the first-order derivative and second-order derivative are both matrices. Although it seems like the process will go through, I still feel something's wrong.
Please give me some suggestions on solving this, thanks!