I am developing in my PhD an new proposed approach using SEM (stochastic expectation maximization algorithm) that fits a multivariate linear mixed model by maximizing the likelihood function to find MLE. The form of the LF is highly complicated due to the large dimension of parameters which requires using numerical techniques based on deriving the Hessian matrix that reach up to size 36X36. Finding this matrix require finding the first and second derivatives of many complicated function such as Kronecker product, determinate, inverse of determinate, inverse of a matrix and so on. My question here related to deriving one of those quantities.
How I differentiate $\large\frac{\partial{(\Sigma \otimes I_n)}}{\partial{\sigma_j}}$
where
- $\Sigma$ is a symmetric positive definite matrix of order $n$
- It is known that $\large\frac{\partial{\Sigma^{-1}}}{\partial{\sigma_j}} = F_j$ , where $\Sigma^{-1}= \sum_{i=1}^{h} \sigma_j F_j$ and $\sigma_j$'s is the non-redundant elements of the precision matrix $\Sigma^{-1}$
- $I_n$ is the identity matrix of order $n$
For an invertible matrix $\Sigma$ we have $$\eqalign{ I &= \Sigma\Sigma^{-1} \cr 0 &= d\Sigma\,\Sigma^{-1} + \Sigma\,d\Sigma^{-1} \cr d\Sigma &= -\Sigma\,\,d\Sigma^{-1}\,\Sigma \cr }$$ Also $$d(\Sigma\otimes I) = (d\Sigma\otimes I)$$ Now substitute the specifics of your matrix into these expressions.