Outliers in the spectrum of iid matrices with bounded rank perturbations

• 成果类型: Article
• 署名作者: Tao, Terence
• 署名单位: University of California System; University of California Los Angeles
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-011-0397-9
• 发表日期: 2013
• 页码: 231-263
• 关键词: sample covariance matrices; circular law; LARGEST EIGENVALUE; singular-values; zeros

摘要

It is known that if one perturbs a large iid random matrix by a bounded rank error, then the majority of the eigenvalues will remain distributed according to the circular law. However, the bounded rank perturbation may also create one or more outlier eigenvalues. We show that if the perturbation is small, then the outlier eigenvalues are created next to the outlier eigenvalues of the bounded rank perturbation; but if the perturbation is large, then many more outliers can be created, and their law is governed by the zeroes of a random Laurent series with Gaussian coefficients. On the other hand, these outliers may be eliminated by enforcing a row sum condition on the final matrix.