Frechet derivative of trace of matrix expression

Question

I am new to the concept of Frechet Derivatives. I have encountered a problem where I am supposed to find the Frechet derivative of $\operatorname{trace}(XAX+AXA^T)$ where my $X,A \in \Bbb R^{n\times n}$ and $X=X^T$.

Answer 1

Let $f(X) = {\rm tr}(XAX+AXA^T)$. Since ${\rm tr}$ is a linear functional, the total derivative of ${\rm tr}$ is ${\rm tr}$ itself, so the product rule gives $$Df(X)(H) = {\rm tr}(HAX+XAH+AHA^T).$$