I want to calculate the gradient of
$$ w^H H F (F^H F)^{-1} F^H H^H w $$
with respect to $ F $, which is complex.
I am basing on this previous answer Derivative of Nested Matrix Quadratic Form that uses differentials to compute the derivative of a similar expression with real matrices. However, I have difficulties in computing the differential when $ (.)^H $ is involved.
For instance, I make these changes: $ x = F^H H^H w $ and $ Z = F^H F $. Then, I obtain $ dx = 0 $ and $ dZ = F^H dF $.
Is it correct that $ dx = 0 $ or should I consider approaching the problem from a different perspective? Thank you!
As you suggested, define the variables $$\eqalign{ x &= F^HH^Hw &\implies x^H = w^HHF \cr Z &= F^HF &\implies Z^{-1}F^H = F^+ {\rm \,\,(pseudoinverse)}\cr }$$ and yes, in the context of Wirtinger derivatives $\,dx=0$.
Write the function in terms of these new variables. Then find its differential and gradient. $$\eqalign{ \phi &= x^HZ^{-1}x \cr d\phi &= dx^HZ^{-1}x + x^HdZ^{-1}x \cr &= dx^HZ^{-1}x - x^HZ^{-1}dZ\,Z^{-1}x \cr &= (w^HH\,dF)Z^{-1}x - x^HZ^{-1}(F^HdF)\,Z^{-1}x \cr &= \Big(Z^{-1}xw^HH - Z^{-1}xx^HZ^{-1}F^H\Big)^T:dF \cr &= \Big(Z^{-1}F^HH^Hww^HH - Z^{-1}F^HH^Hww^HHFZ^{-1}F^H\Big)^T:dF \cr &= \Big(F^+H^Hww^HH - F^+H^Hww^HHFF^+\Big)^T:dF \cr &= \Big((F^+H^Hww^HH)\,(I - FF^+)\Big)^T:dF \cr \frac{\partial\phi}{\partial F} &= (I - FF^+)^T (F^+H^Hww^HH)^T \cr }$$ where a colon was used in some steps as a convenient product notation for the trace, i.e. $$A:B = {\rm Tr}(A^TB)$$