I was reading a paper and I encountered an expression that I think is wrong, but I am not completely sure. The expression is: $$\mbox{tr}\left[\dfrac{\partial[|\mbox{tr}(AB)|^2]}{\partial B} C\right] = \dfrac{\partial[\mbox{tr}(AB)\mbox{tr}(AB)^\ast]}{\partial B_{ji}}C_{ji} = A_{ij}\mbox{tr}(AB)^\ast C_{ji} = \mbox{tr}(AC)\mbox{tr}(AB)^\ast$$
My problem is in the very first passage (if that is correct, than the rest is also correct. The paper uses the Einstein convention for summations and use the Wirtinger calculus for the derivatives.
My problem with the first passage is that I think it should be $B_{ij}$ instead of $B_{ji}$ but I am afraid I must be confused with the notation.
Can somebody please take a look and tell me if this is in fact wrong or the notation is tricking me?
Thank you very much.
Since the author is using the numerator layout, $$ \left(\frac{\partial y}{\partial\mathbf{X}}\right)_{ij}=\left(\frac{\partial y}{\partial X_{ji}}\right) $$ So $$ \operatorname{tr}\left[\frac{\partial[\lvert\operatorname{tr}(AB)\rvert^2]}{\partial B}C\right]= \left(\frac{\partial[\lvert\operatorname{tr}(AB)\rvert^2]}{\partial B}\right)_{ij}C_{ji}= \left(\frac{\partial[\lvert\operatorname{tr}(AB)\rvert^2]}{\partial B_{ij}}\right)C_{ji} $$ is correct.