Let us fix dimension $n$. Consider the $n \times n$ matrix \begin{equation} S_n=\begin{bmatrix} 1 & z & z & \cdots & z \\ \bar{z} & 1 & z & \cdots & z\\ \bar{z} & \bar{z} & 1 & \cdots & z\\ \vdots &\vdots &\vdots & \ddots &\vdots \\ \bar{z} & \bar{z} & \bar{z} & \cdots & 1 \end{bmatrix}. \end{equation} When is this matrix positive?
Partial answer, if $z$ be real, say $r$, then it is easy to see that $0 \le r \le 1$. In general if $z=r e^{\imath\theta}$, then $0 \le r \le 1$ is a necessary condition, not sufficient.
The above criteria, if exists, is dimension $n$ dependent. What is the condition such that $S_n$ is positive for all $n$? I tried to construct the corresponding measure. But it seems very bizarre. Probably I am making some mistake. Please help.