## 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 |
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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 | |

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 Sep 2011 → 30 Sep 2011 |

### Other

Other | 2011 49th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2011 |
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Country | United States |

City | Monticello, IL |

Period | 28/09/11 → 30/09/11 |