Spectral norm of products of random and deterministic matrices

TL;DR

This paper establishes sharp bounds on the spectral norm of products of random matrices with independent entries and deterministic matrices, showing they behave similarly to fully random matrices, under certain moment conditions.

Contribution

It provides new bounds on the spectral norm of such matrix products and implications for the smallest singular value, extending prior work.

Findings

Spectral norm of M=BA is bounded by rac{m}{} + rac{n}{} under (4+epsilon)-th moment.

Spectral norm bounds are sharp and comparable to fully random matrices.

Smallest singular value is bounded below by rac{m}{} - rac{n-1}{} with high probability.

Abstract

We study the spectral norm of matrices M that can be factored as M=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 by n matrix M 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 M. Rudelson and the author implies that the smallest singular value of a random m times n matrix with i.i.d. mean zero entries and bounded (4+epsilon)-th moment is bounded below by \sqrt{m} - \sqrt{n-1} with high probability.