Asymptotic results on generalized Vandermonde matrices and their extreme eigenvalues

This paper examines various statistical distributions in connection with random N × N Vandermonde matrices and their generalization to d-dimensional phase distributions. Upper and lower bound asymptotics for the maximum eigenvalue are found to be O(log N^d) and O(log N^d/ log log N^d) respectively. The behavior of the minimum eigenvalue is considered by studying the behavior of the maximum eigenvalue of the inverse matrix. In particular, we prove that the minimum eigenvalue λ<inf>1</inf> is shown to be at most O(exp(−√NW^∗<inf>N</inf>)) where W^∗<inf>N</inf> is a positive random variable converging weakly to a random variable constructed from a realization of the Brownian Bridge on [0, 2π). Additional results for (V^∗V)⁻¹, a trace log formula for V^∗V, as well as a some numerical examinations of the size of the atom at 0 for the random Vandermonde eigenvalue distribution are also presented.