Deviation Inequalities on the Spectral Norm of Products of Random and Deterministic Matrices

Guozheng Dai, Zhonggen Su, Hanchao Wang

Published 2023-07-06Version 1

We study the spectral norm of matrices $BA$, where $A$ is a random matrix with independent mean zero subexponential entries, and $B$ is a fixed matrix. We show that the spectral norm of such an $m\times n$ matrix $BA$ exceeds $\sqrt{n}+\sqrt{m}$ with an exponential decay probability. Applying this result, we prove an estimate of the smallest singular value of a random subexponential matrix using an argument in the previous work of Rudelson and Vershynin.