Matrix expression for the derivative of matrix to a power

Question

Let us pick an example to be more clear:

Let $\phi: M(n) \to M(n)$ defined by $\phi(A) = A^2$

I know it's derivative is given by $\phi'(A)\cdot H = HA+AH$, but i was wondering, how to write it's derivative as a matrix expresion? I just wanted some example of it to get the idea, but i couldn't find anywhere.

Answer 1

The jacobian matrix of the application is a $n^2 \times n^2$ matrix. You have to choose a basis (for instance $(E_{ij})_{i,j\in[1,n]}$) and flatten it ($e_1 = E_{1,1}, e_2 = E_{1,2},... e_n = E_{1,n},e_{n+1} = E_{2,1},...e_{n^2} = E_{n,n}$) and write the associated matrix.

It's painful to write and use in calculations.