### Abstract

Consider a truncated circular unitary matrix which is a p_{n} by p_{n} submatrix of an n by n circular unitary matrix by deleting the last n−p_{n} columns and rows. Jiang and Qi [11] proved that the maximum absolute value of the eigenvalues (known as spectral radius) of the truncated matrix, after properly normalized, converges in distribution to the Gumbel distribution if p_{n}/n is bounded away from 0 and 1. In this paper we investigate the limiting distribution of the spectral radius under one of the following four conditions: (1). p_{n}→∞ and p_{n}/n→0 as n→∞; (2). (n−p_{n})/n→0 and (n−p_{n})/(logn)^{3}→∞ as n→∞; (3). n−p_{n}→∞ and (n−p_{n})/logn→0 as n→∞ and (4). n−p_{n}=k≥1 is a fixed integer. We prove that the spectral radius converges in distribution to the Gumbel distribution under the first three conditions and to a reversed Weibull distribution under the fourth condition.

Original language | English (US) |
---|---|

Pages (from-to) | 536-554 |

Number of pages | 19 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 458 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2018 |

### Keywords

- Circular unitary matrix
- Eigenvalue
- Extreme value
- Limiting distribution
- Spectral radius