I am trying to understand the complex gradient fundamentals here.
For example in [1, Example A.3(3)]--also attached the snapshot below for the convenience
.
How to arrive at the complex gradient of $g(z) = \kappa + a^*z + z^*b + z^* C z \ ,$ where $a,b,z \in \mathbb{C}^{n \times 1}$, and $C \in M_{n,n} \left(\mathbb{C}\right)$ (not necessarily it is Hermitian matrix) as $\frac{\partial g(z) }{\partial z^*} = b + C z$ ?
I think the rules of complex-valued gradients would/should be different from the real-valued gradients since $${\rm Tr}\left(X^* Y\right) \neq {\rm Tr}\left(Y^* X \right)$$ or $$x^* y \neq y^* x$$
My question is, should I treat the conjugate component as a "constant"?
In other words, if
\begin{align}
g(z) &= \kappa + a^*z + z^*b + z^* C z \\
& = \kappa + \overline{a} : z + \overline{z}:b + \overline{z} : C z \ ,
\end{align}
where $\overline{z}$ denotes the complex conjugate of $z$, and $x : y = {\rm Tr} \left( x^T y \right) = x^T y$.
So, if I want to compute the derivate of $g(z)$ w.r.t. $z^*$ and treat $z$ as a "constant" (if my hypothesis is correct), then the differential will read \begin{align} dg(z) & = \underbrace{\overline{a} : dz}_{0} + \underbrace{ d\overline{z}:b }_{= \ b \ : \ d\overline{z} } + \underbrace{d\overline{z} : C z}_{ = \ C z \ : \ d\overline{z}} + \underbrace{\overline{z} : C dz}_{0}\ \\ &= b \ : \ d\overline{z} + C z \ : \ d\overline{z} \\ &= \left( b + C z \right) : \ d\overline{z} \ , \end{align} then, the gradient is $$\frac{\partial g(z) }{\partial z^*} = b + C z \ .$$
Am I on the correct path? Thank you for your patience.
1 Sayed, Ali H., Adaptation, learning, and optimization over networks, Found. Trends Mach. Learn. 7, No. 4-5, 311-801 (2014). ZBL1315.68212.