Does this System of Complex Variables Has Solutions?

Question

Find all the complex vectors $\mathbf{x}=[x_1,\ldots,x_n]^\top$ and $\mathbf{y}=[y_1,\ldots,y_n]^\top$ in $\mathbb{C}^n$ such that $$ \sum_{i\in S}x_i\bar{y_i}=1,\text{ for all } S\subset\{1,\ldots,n\}\text{ and } S\neq\emptyset, $$ where $\bar{\cdot}$ denotes the conjugate. I think I can find a special solutions. Can I find a general formula of the solutions?

Answer 1

If $n > 1$, we cannot have

$$\sum_{i \in S} x_i \overline{y}_i = 1\tag{$\ast$}$$

for all nonempty $S\subset \{1,2,\dotsc,n\}$. Looking at the singleton sets $S = \{i\}$, it follows that a necessary condition is $x_i \overline{y}_i = 1$ for all $1 \leqslant i \leqslant n$, and then we have

$$\sum_{i\in S} x_i \overline{y}_1 = \operatorname{card} S.$$

Thus, for $n > 1$ the system $(\ast)$ is inconsistent.