Let $a_1,\dots,a_n\in\mathbb{C}$ be given. Consider minimizing the function of $z)$ $$ f(z_1,\dots,z_n)=\left|\sum_{j=1}^n z_j a_j\right|^2 $$ under the constraint $\left|\sum_{j=1}^n z_j \right|^2=1$.
This question is somewhat trivial: the comments in Minimizing weighted complex sum under constraint show that if any two of the $a_k$'s are distinct, then $f$ attains zero.
I wonder, what if $$ \tilde{f}(z_1,\dots,z_n)=\sum_{j=1}^n \sum_{k=1}^n z_j\overline{z_k} \beta_{jk}, $$ where $\beta_{jk}\in\mathbb{C}$ are given, $\beta_{jk}=\overline{\beta_{kj}}$, with $\left|\sum_{j=1}^n z_j \right|^2=1$? I guess this is like asking an explicit formula of an eigenvector of $(\beta_{jk})$ corresponding to the smallest eigenvalue, so asking too much?