Q:

differential (Jacobi Matrix) of $f(A)=A^2$ where $A$ is a matrix - check my answer

I just want a quick verification that what I did here is correct:
let $f(A)=A^2$ where $A$ is a n by n matrix with real entries.
then $$D_f(A)=\lim_{t \to 0} \frac{f(A+tA)-f(A)}{t} = \lim_{t \to 0} \frac{(A+tA)^2-A^2}{t} = \lim_{t \to 0} \frac{A^2+AtA+tAA+t^2A^2-A^2}{t}=\lim_{t \to 0} \frac{2tA^2+t^2A^2}{t} =\lim_{t \to 0} 2A^2+tA^2 =2A^2$$
so overall
$$D_f(A)=2A^2$$
is this correct?

A:

It depends on the definition of the differential. A definition if a map $L$ which maps any $A$ to a linear map $L(A)$ such that 
$$\lim_{\lVert H\lVert\to 0}\frac{f(A+H)-f(A)-L(A)(H)}{\lVert H\rVert}=0.$$
With this definition, we find $Df(A)(H)=AH+HA$.
Here you computed correctly the directional derivative at $A$ in the direction $A$, but we also have to consider the other directions.

