Given matrices $A$ and $B$ and scalar field $$g(X) := \operatorname{tr}(AXB)$$ compute the gradient $\nabla_X g$.
I do not want a solution, i just got a question about an equation of a derivative of a trace?
I read in a book that i can apply this rule :
and my question was if i can write $\frac{ \partial} {\partial X} tr(AXB)$ = $tr(\frac{ \partial AXB} {\partial X})$ ?
If i compute now the derivative of it then i get $tr(B^T⊗A)$ ?
You have a combination of linear functions so the derivative will be the same function. Using the trace cyclic property: $tr (AXB) = tr (BAX) = tr (CX)$, we can write for the derivative map $d[ tr(CX)] = tr(C\circ id)$ -- a composition of maps with the inner-most identity map, (and I'd like to hear a critique and an alternative to that notation), expectedly independent of $X$. Evaluated at an argument $H$ we get $d[ tr(CX)](H) = tr(CH)$