Summary: Let $K, M \in \Bbb N$ with $K < M$, and define a square $K\times K$ Vandermonde matrix $A = A (\tau,\vec n)$ with nodes on the unit circle: $A_{p,q} = \exp (- j2\pi pn_q\tau/K)$; $p,q = 0,1,\dots,K-1$, where $n_q\in \{0,1,\dots,M-1\}$ and $n_0 <n_1<\dots < n_{K-1}$. Such matrices arise in some types of interpolation problems. In this paper, necessary and sufficient conditions are presented on the vector $\vec n$ so that a value of $\tau\in \Bbb R$ can be found to achieve perfect conditioning of $A$. A simple test to check the condition is derived and the corresponding value of $\tau$ is found.

15A12, 65F35

vandermonde matrix, condition number, perfect conditioning