On perfect conditioning of Vandermonde matrices on the unit circle

Abstract

Let K,M ∈ N with K<M, and define a square K × K Vandermonde matrix A = A (τ, )
with nodes on the unit circle: A_p,q = exp (−j2πpn_qτ /K) ; p, q = 0, 1, ..., K − 1, where
n_q ∈ {0, 1, ...,M − 1} and n₀ < n₁ < .... < n_K−1. Such matrices arise in some types of interpolation
problems. In this paper, necessary and sufficient conditions are presented on the vector −→n so that
a value of τ ∈ R can be found to achieve perfect conditioning of A. A simpl e test to check the
condition is derived and the corresponding value of τ is found.