Consider a matrix $A$ and its characteristic equation \begin{equation} \Phi(A) = \sum_{k=0}^n c_k(A) \lambda^{n-k}. \end{equation} $c_k$ can be worked out in many ways but one is a recursive method derived from the Fadddeev-LeVerrier algorithm: \begin{equation} c_k(A) = -\frac{1}{k}\sum_{i=0}^{k-1} \operatorname{Tr}[A^{k-i}]c_i(A). \end{equation}
I would like to Taylor expand $c_k$ around a small perturbation of $A$.
I have read here (https://mathoverflow.net/questions/139643/taylor-expansion-of-a-function-of-a-matrix) that the formula for this to first order is \begin{equation} f(A+\epsilon B) = f(A) + \epsilon f'_A(B) + \mathcal{O}(\epsilon^2) \end{equation} but I do not know what the notation `$f'_A(B)$' means.
I have tried to apply this to $c_k$ and, guessing that we can use the product rule, get to \begin{equation} c_k(A+\epsilon B) \simeq c_k(A) -\frac{\epsilon}{k} \sum_{i=0}^{k-1} \operatorname{Tr}[A^{k-i}]'_A(B) c_i(A) + \operatorname{Tr}[A^{k-i}]c_i(A)'_A(B). \end{equation} I have looked online for notes on how to do this but can't find anything.
Questions:
Do you have a reference for the equation Taylor expansion equation that explains it?
Can you help finish the Taylor expansion of $c_k(A)$?
Thank you.
Let $H\in M_n$ be a small matrix. Then $c_k(A+H)\approx c_k(A)+Dc_{k}(H)$ where the last term is given by the recurrence formula:
$Dc_0(H)=0;k\geq 1\implies Dc_{k}(H)=-1/k\sum_{i=0}^{k-1}((k-i)tr(A^{k-i-1}H)c_i(A)+tr(A^{k-i})Dc_{i}(H))$.