Asymptotic results on generalized Vandermonde matrices and their extreme eigenvalues

Gabriel H. Tucci; Philip A. Whiting

doi:10.1109/Allerton.2011.6120389

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 proceeding › Conference proceeding contribution › peer-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 language	English
Title of host publication	2011 49th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2011
Place of Publication	Piscataway, NJ
Publisher	Institute of Electrical and Electronics Engineers (IEEE)
Pages	1816-1823
Number of pages	8
ISBN (Electronic)	9781457718168, 9781457718182
ISBN (Print)	9781457718175
DOIs	https://doi.org/10.1109/Allerton.2011.6120389
Publication status	Published - 2011
Externally published	Yes
Event	2011 49th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2011 - Monticello, IL, United States Duration: 28 Sept 2011 → 30 Sept 2011

Other

Other	2011 49th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2011
Country/Territory	United States
City	Monticello, IL
Period	28/09/11 → 30/09/11

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10.1109/Allerton.2011.6120389

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title = "Asymptotic results on generalized Vandermonde matrices and their extreme eigenvalues",

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.",

author = "Tucci, {Gabriel H.} and Whiting, {Philip A.}",

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doi = "10.1109/Allerton.2011.6120389",

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booktitle = "2011 49th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2011",

publisher = "Institute of Electrical and Electronics Engineers (IEEE)",

address = "United States",

note = "2011 49th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2011 ; Conference date: 28-09-2011 Through 30-09-2011",

}

Tucci, GH & Whiting, PA 2011, Asymptotic results on generalized Vandermonde matrices and their extreme eigenvalues. in 2011 49th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2011., 6120389, Institute of Electrical and Electronics Engineers (IEEE), Piscataway, NJ, pp. 1816-1823, 2011 49th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2011, Monticello, IL, United States, 28/09/11. https://doi.org/10.1109/Allerton.2011.6120389

Asymptotic results on generalized Vandermonde matrices and their extreme eigenvalues. / Tucci, Gabriel H.; Whiting, Philip A.
2011 49th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2011. Piscataway, NJ: Institute of Electrical and Electronics Engineers (IEEE), 2011. p. 1816-1823 6120389.

Research output: Chapter in Book/Report/Conference proceeding › Conference proceeding contribution › peer-review

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AU - Tucci, Gabriel H.

AU - Whiting, Philip A.

PY - 2011

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N2 - 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.

AB - 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.

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BT - 2011 49th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2011

PB - Institute of Electrical and Electronics Engineers (IEEE)

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ER -

Asymptotic results on generalized Vandermonde matrices and their extreme eigenvalues

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