Spectral norm of products of random and deterministic matrices

• 成果类型: Article
• 署名作者: Vershynin, Roman
• 署名单位: University of Michigan System; University of Michigan
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-010-0281-z
• 发表日期: 2011
• 页码: 471-509
• 关键词: smallest singular-value; LARGEST EIGENVALUE; INVERTIBILITY; limit

摘要

We study the spectral norm of matrices W that can be factored as W = BA, where A is a random matrix with independent mean zero entries and B is a fixed matrix. Under the (4 + epsilon)th moment assumption on the entries of A, we show that the spectral norm of such an m x n matrix W is bounded by root m + root n, which is sharp. In other words, in regard to the spectral norm, products of random and deterministic matrices behave similarly to random matrices with independent entries. This result along with the previous work of Rudelson and the author implies that the smallest singular value of a random mxn matrix with i.i.d. mean zero entries and bounded (4+epsilon)th moment is bounded below by root m - root n - 1 with high probability.