I am trying to solve the following problem.
Find $\mathbf{q}\in\mathbb{S}^{2}$(unit sphere) such that $$\arg\max_{\mathbf{q}\in\mathbb{S}^{2}}|\mathbf{E}\cdot(\overline{\underline{\mathbf{G}}\cdot \mathbf{q}})|$$ where $\mathbf{E}\in\mathbb{C}^{3}$ and $\underline{\mathbf{G}}\in\mathbb{C}^{3\times3}$ are given data.
Here $\overline{\underline{\mathbf{G}}}$ is complex conjugate of $\underline{\mathbf{G}}$. The under line means the matrix
My attempt
For $f(\mathbf{q})=|\mathbf{E}\cdot(\overline{\underline{\mathbf{G}}\cdot \mathbf{q}})|$, find $\mathbf{q}$ such that $\nabla_{\mathbf{q}} f(\mathbf{q})=0$.
My question is
(1) is this approach correct?
(2) Since $\mathbf{E}\in\mathbb{C}^{3}$ and $\underline{\mathbf{G}}\in\mathbb{C}^{3\times3}$ are complex values, should I have to find $\mathbf{q}$ such that $Real(\nabla_{\mathbf{q}} f(\mathbf{q}))=0$ and $Image(\nabla_{\mathbf{q}} f(\mathbf{q}))=0$
(3) Can you provide a better approach to me if my attempt is wrong?
Thank you so much.