Gradient of quadratic form

Question

I am new to linear algebra. I appreciate if somebody helps me to solve this problem. I have a function $f(v)=hvv^Hh^H$, where $h\in \mathbf{C}^{1\times N}$ is a constant vector, $v\in \mathbf{C}^{N\times 1}$ is variable vector, and $H$ denotes Hermitian transpose. I need to calculate $\nabla f(v)$

Answer 1

For consistency, use column vectors so that both ${\,h,v\in\mathbb C}^{N\times 1}$

Then consider the complex scalar $$\phi = h^Tv$$ in terms of which your (real) function is $$f= |\phi|^2 = \phi^*\phi$$ Since $\phi$ depends solely upon $v$ (and $\phi^*$ upon $v^*$) we can easily find the gradient as $$\eqalign{ df &= \phi^*d\phi = (\phi^*h)^Tdv \cr \frac{\partial f}{\partial v} &= \phi^*h \cr }$$ Taking the complex conjugate yields $$ \frac{\partial f}{\partial v^*} = \phi h^* \implies \frac{\partial f}{\partial v^H} = \phi h^H $$