Matrix by matrix differentiation question

Question

$A,B,C$ are matrices, and $C=AB$.
$l$ is a scalar, and $l=f(C)$.

If $\dfrac{\partial l}{\partial C}$ is known, what is $\dfrac{\partial l}{\partial A}$?

Answer 1

Use the Frobenius Inner Product to write the differential of the function, then perform a change of variables $$\eqalign{ dl &= \frac{\partial l}{\partial C}:dC \cr &= \frac{\partial l}{\partial C}:dA\,B \cr &= \frac{\partial l}{\partial C}\,B^T:dA \cr\cr \frac{\partial l}{\partial A} &= \frac{\partial l}{\partial C}\,B^T \cr }$$