Dear François, I'm indeed interested in the script you mentioned in your answer, could you please put the link to the script here? That will be very helpful. Thank you in advance.

I think The Bridge talks about their "book" whose title is "Financial Modelling with Jump Processes"

if $c$ is a function of time too, just consider $$Z_t=\exp(-\int_0^tc(s,X_s))ds$$ then you have $dZ_t=-Z_tc(s,X_s)dt$ and the same reasoning applies.

which page are you looking at?

I don't see how to make further progress, could you give the reference of the book you've mentioned?

Do you have additional information on the $\alpha_i$ or they are of general form?

Sorry for posting this as an answer, I cannot leave comment. Are you sure your matrix is $$[A(k)]_{j\,l}= -\frac{e^{ik|y_j-y_l|}}{4\pi|y_j-y_l|}\, j\neq l$$ and not $$[A(k)]_{j\,l}= -\frac{e^{ik|y_j-y_l|}-1}{4\pi|y_j-y_l|}\, j\neq l$$ If you want to extend some properties of the derivative of a matrix to its eigenvalue, the eigenvectors have to be independent of the derivation variable.

I edited the first answer to take into account your comment