In this paper I'm reading, they say that if
$\Psi(A) = A^2$ then the derivative $D\Psi(A)$ is defined by $D\Psi(A)(X) = AX + XA.$
I'm a bit confused by that line since I know the differential is given by $d(A^2) = A\, dA + dA\, A$.
But to me this looks a little different. I don't see how the $X$ would pop up in the derivative $AX+XA$ in place of $dA$ when we go from a differential to a derivative.
Can someone explain where I'm going wrong and clarify?
You have for every matrix $H$:
$\Psi(A+H)=(A+H)^{2}=(A+H)(A+H)=A^{2}+AH+HA+H^{2}=\Psi(A)+AH+HA+H^{2}$,
where the expression $AH+HA$ is linear in $H$. Hence:
$\lim_{H\to 0}\dfrac{\Psi(A+H)-\Psi(A)-(AH+HA)}{||H||}=\lim_{H\to 0}\dfrac{H^{2}}{||H||}=0.$
Therefore, $\Psi$ is differentiable at $A$ and $d\Psi_{A}(H)=AH+HA$ for every $H$.