Asymptotic results on generalized Vandermonde matrices and their extreme eigenvalues

Gabriel H. Tucci*, Philip A. Whiting

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference proceeding contributionpeer-review

3 Citations (Scopus)

Abstract

This paper examines various statistical distributions in connection with random N x 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 1 is shown to be at most O(exp(-√NW N*)) where W N* 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.

Original languageEnglish
Title of host publication2011 49th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2011
Place of PublicationPiscataway, NJ
PublisherInstitute of Electrical and Electronics Engineers (IEEE)
Pages1816-1823
Number of pages8
ISBN (Electronic)9781457718168, 9781457718182
ISBN (Print)9781457718175
DOIs
Publication statusPublished - 2011
Externally publishedYes
Event2011 49th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2011 - Monticello, IL, United States
Duration: 28 Sep 201130 Sep 2011

Other

Other2011 49th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2011
CountryUnited States
CityMonticello, IL
Period28/09/1130/09/11

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