I want to take the derivative of $$\frac{\partial}{\partial W}(WS S^HW^H)$$ Where $W\in \mathbb C^{N\times M}$ and $S\in \mathbb C^{M\times M}$ symmetric matrix. My question, Is it straightforward derivative of quadratic form that should look like this $$2SS^HW$$
keeping in mind that all matrices are complex valued?
I appreciate your comments on it!
The derivative of the sesquilinear map
$$\Phi : (U,V) \mapsto USS^HV^H$$ is $$d\Phi(U,V).(A,B) = A SS^H V^H + USS^HB^H.$$ Hence $$\frac{\partial}{\partial W}(WS S^HW^H).A = A SS^H W^H + WSS^HA^H.$$ As $S$ is supposed to be hermitian, $A SS^H W^H = WSS^HA^H$ and therefore $$\frac{\partial}{\partial W}(WS S^HW^H).A = 2 A SS^H W^H = 2WSS^HA^H$$