ON THE REAL DAVIES' CONJECTURE

• 成果类型: Article
• 署名作者: Jain, Vishesh; Sah, Ashwin; Sawhney, Mehtaab
• 署名单位: Stanford University; Massachusetts Institute of Technology (MIT)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/21-AOP1522
• 发表日期: 2021
• 页码: 3011-3031
• 关键词: Matrices

摘要

We show that every matrix A is an element of R-n(xn) is, at least, delta parallel to A parallel to-close to a real matrix A + E is an element of R-nxn whose eigenvectors have condition number, at most, (O) over tilde (n)(delta(-1)). In fact, we prove that, with high probability, taking E to be a sufficiently small multiple of an i.i.d. real sub-Gaussian matrix of bounded density suffices. This essentially confirms a speculation of Davies and of Banks, Kulkarni, Mukherjee and Srivastava, who recently proved such a result for i.i.d. complex Gaussian matrices. Along the way we also prove nonasymptotic estimates on the minimum possible distance between any two eigenvalues of a random matrix whose entries have arbitrary means; this part of our paper may be of independent interest.