Spectral norm of products of random and deterministic matrices

Abstract

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 + e)th moment assumption on the entries of A, we show that the spectral norm of such an m × n matrix W is bounded by $${\sqrt{m} + \sqrt{n}}$$ , 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 m × n matrix with i.i.d. mean zero entries and bounded (4 + e)th moment is bounded below by $${\sqrt{m} - \sqrt{n-1}}$$ with high probability.