Let $M_{n}(\mathbb{R})$ the set of $n\times n$ matrix and $f\colon M_{n}(\mathbb{R})\rightarrow M_{n}(\mathbb{R})$ such that $$f(A)=A^{3}+A^{2}.$$ Find $Df_{A_{0}}(H)$, with $H\in M_{n}(\mathbb{R})$.
I was trying to identify each matrix of $M_{n}(\mathbb{R})$ as a vector of $\mathbb{R}^{n^{2}}$ to find the partial derivates of $f$ and then find the Jacobian matrix, but is inviable. There is some result to use here? How it word the differential of matrix?
Obs: Sorry about my english.
$$f'(A)(H)=HAA+AHA+AAH\ +\ HA+AH\ .$$ For this consider algebraically $$ f(A+H)-f(A)=(A+H)(A+H)(A+H)+(A+H)(A+H)-AAA-AA\ , $$ and isolate the linear part in $H$. Alternatively, consider
the trilinear form $t:(A_1,A_2,A_3)\to A_1A_2A_3$, respectively
the bilnear form $b:(A_1,A_2)\to A_1A_2$, and use that the derivative of a linear function at any point is the linear function itself.