I'm a beginner and I have a question. I'm sorry for my bad english.
Let $A$ be an invertible n by n matrix, and let $F$ be a function defined on $M_n(C)$ by $F(X) = X^2 - A$.
I would like to know how we can calculate $DF(X)(H)$ the differential of $F$ at the point $X \in M_n(C)$ for an increase $H \in M_n(C)$, using a Taylor development at the order 1.
Could someone help me ? Thank you in advance.
In this case, the best way to calculate the directional derivative is straight by definition.
$\begin{align}DF(X)H&=\lim_{t\to0}\frac{F(X+tH)-F(X)}{t}\\&=\lim_{t\to0}\frac{tHX+tXH+t^2H^2}{t}\\&=HX+XH.\end{align}$