I have the following function which I need to find the derivative of $$L=trace(\Sigma K^{-1})$$ where $K$ is a function of $\theta$ and $\Sigma$ is constant.
If I'm correct what I need to do to find $\frac{\partial L}{\partial \theta}$ is $\frac{\partial L}{\partial K}\times\frac{\partial K}{\partial \theta}$. Correct me if I am wrong.
Question is what is $\frac{\partial L}{\partial K}$ ?
This is not to hard if you write it out in components. Since $ L = \text{Tr}(A_{ij}) = \sum_i A_{ii} $, $ \frac{dL}{d\theta} = \sum_i \frac{dA_{ii}}{d\theta} = \text{Tr} (\frac{dA_{ij}}{dt}) $. So it is not quite a chain rule with matrix multiplication the way you wrote it out.
For inside the trace, you can also use the $ \frac{d}{d\theta} K^{-1} = -K^{-1} \frac{dK}{d\theta} K^{-1} $, which you can derived from $ K K^{-1} = I $.