Take the matrix $\Sigma\in\mathbb{R}^{n\times n}$ and the function $f:\mathbb{R^n}\rightarrow\mathbb{R}$ in $C^2$.
1) How can I compute the matrix derivation
$$\frac{\partial (tr\left(\Sigma\Sigma^TD^2f\right))}{\partial{\Sigma}}$$
where $tr$ is the trace operator and $D^2f$ the Hessian matrix?
2) How does it simplify, if I choose $\Sigma$ to be a diagonal matrix?
My aim is to see if there can be found a form of the derivative in terms of the Laplacian $\Delta{f}$ and the gradient $\nabla{f}$.
You may be able to use a standard result regarding matrix derivatives:
$$ \frac{\partial Tr(X^TAX)}{\partial X} = (A + A^T)X $$ Separately, the Trace has the property (for appropriate matrix dimensions):
$$ Tr(AB) = Tr(BA) $$
Now, denoting the Hessian of $f$ as $H$, We obtain:
$$ \frac{\partial Tr(\Sigma \Sigma^T H)}{\partial \Sigma } = \frac{\partial Tr(\Sigma^T H\Sigma )}{\partial \Sigma } = (H + H^T)\Sigma = 2H\Sigma $$
Where the last equality follows since the Hessian is symmetric.
For the second part of your question, it should be fairly obvious that if $\Sigma$ is diagonal, then $2H\Sigma $ becomes the Hessian $H$ with each column $h_i$ scaled by a factor of $2\Sigma_{ii}$.