Suppose F is a field, and $M_n(F)$ is a matrix algebra over F, denoted by V. If $\phi \in L(V)$, the set of all linear map between V, and $\phi$ satisfies Leibeniz's law: $\phi(AB) = \phi(A)B+A\phi(B)$ for any A,B$\in M_n(F)$, then we call such $\phi$ is a derivation.
Now, I noted that when fixing a matrix A $\in M_n(F)$, then the linear map $ad_A(B)=AB-BA$ is a derivation. Could everybody gives me some suggestions of finding all the derivation of the matrix algebra? I have a intuition that the rest of the derivation can be derived by $ad_A$ .
Like, suppose $\mathcal{D}$ is a derivation of the matrix algebra, then $$\mathcal{D}(ad_A(B))=\mathcal{D}(AB-BA)=\mathcal{D}(A)B+A\mathcal{D}(B)-\mathcal{D}(B)A-B\mathcal{D}(A)$$ $$=ad_A(\mathcal{D}(B))-ad_B(\mathcal{D}(A))$$ However, I have no idea how to go further to indentify the structure of derivation of matrix algebra. Thank you for your help in advance!
