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 $\left|\sum_{j=1}^n z_j \right|^2=1$.
What would $(z_1,\dots,z_n)$ that minimizes $f$ be?
Naturally, I considered minimizing $\sum_{j=1}^n\sum_{k=1}^n z_j\overline{z_k} a_j\overline{a_k}$ under $\sum_{j=1}^n\sum_{k=1}^n z_j\overline{z_k} =1$ but each $z_j$, $a_j$ are not real and I do not know how to proceed.