I have the following function
$$f=\text{Tr(A}^T\text{SA}),$$
where $A$ is a matrix, $S$ is given by
$$S_{ij} = \exp(-\frac{(x_i-x_j)^2}{2\sigma^2})$$
Here is my attempt to calculate the derivative of $f$ w.r.t $X$
We have
$$\begin{align*} f &= \text{Tr}(AA^TS) \\ &= AA^T:S \\ \implies df &=AA^T:dS \\ &= AA^T:(\frac{-1}{2\sigma^2})S\odot dS \\ &= (\frac{-1}{2\sigma^2}) AA^T:S\odot d(diag(X^TX)1^T + 1(diag(X^TX))^T - 2X^TX) \\ &= (\frac{-1}{2\sigma^2}) AA^T:S\odot (diag(2X^TdX)1^T + 1(diag(2X^TdX))^T - 2X^TdX) \\ &= (\frac{-1}{2\sigma^2})(S\odot AA^T): (diag(2X^TdX)1^T + 1(diag(2X^TdX))^T - 2X^TdX) \\ &= (\frac{-1}{2\sigma^2})[-2X(S\odot AA^T):dX + (S\odot AA^T):(diag(2X^TdX)1^T + 1(diag(2X^TdX))^T)] \\ \end{align*}$$
I am stuck with the two terms involving the diag operator,
Could anyone help me to simplify these terms and obtain the final expression of $\frac{df}{dX}$?
Best regards,