How would I go about calculating the derivative with respect to $x$ of $$Q(x) = x^H A x $$ with $A$ a real matrix (not necessarily symmetric) and $x$ a complex valued vector? Here $(\cdot)^H$ denotes the conjugate transpose.
EDIT: This question came up in the context of the maximization problem:
$$ \text{arg}\max_x \, x^H A x \qquad \text{s.t.} \quad x^Hx = 1$$
I think for solving this I need to solve $ \nabla_x \, Q (x) \stackrel{!}{=} 0$, so what I am looking for is the complex gradient vector.
Am I correct in assuming (following the Matrix Cookbook and using that $A$ is real) that:
$$ \nabla_x \, Q(x) = \frac{\partial Q}{\partial Re(x)} + i \frac{\partial Q}{\partial Im(x)} = \, \dots \, = 2 A x$$
where I used that $Q(x) = (Re(x) - i\, Im(x))^T A \,(Re(x) + i\,Im(x)) $?
EDIT2: The above seems to only hold if $x^H A \, x$ is real (additional question: for which matrices $A$ would that be the case? Maybe when I can decompose $A$ into $A = B^TB$ or respectively $B^HB$ for complex A?).