On the smallest singular value of the product of random and deterministic matrices

Abstract

Let $A=(a_{ij})$ be an $n\times n$ real-valued random matrix with independent, mean-zero, variance-one entries whose fourth moments are uniformly at most $K$. Suppose that there exists $\kappa \in (0,1)$ such that the entries of $A$ satisfy $$\underset{i,j}{\max }\underset{u\in \mathbb{R}}{\sup }\mathbb{P}(|a_{ij}-u|<1)\le \kappa .$$ We prove that there are constants $c,C>0$, depending only on $K$ and $\kappa$, such that for every fixed invertible $n\times n$ matrix $M$ and every $\epsilon \ge 0$, $$\mathbb{P}!\left(s_{\min }(MA)\le \frac{\epsilon }{\Vert M^{-1}{\Vert }_{\mathrm{HS}}}\right)\le C\epsilon +e^{-cn}.$$ In the Gaussian case, we also show that the above estimate is sharp in the sense that $\mathbb{E}[s_{\min }(MA)]\asymp \Vert M^{-1}{\Vert }_{\mathrm{HS}}^{-1}.$

20 pages