The Smallest Singular Value of a Shifted Random Matrix

Abstract

Let $R_n$ be a $n\times n$ random matrix with i.i.d. subgaussian entries. Let $M$ be a $n\times n$ deterministic matrix with norm $\Vert M\Vert \le n^{\gamma }$ where $1/2<\gamma <1$. The goal of this paper is to give a general estimate of the smallest singular value of the sum $R_n+M$, which improves an earlier result of Tao and Vu.