A remark on the smallest singular value of powers of Gaussian matrices

Abstract

Let $n,k\ge 1$ and let $G$ be the $n\times n$ random matrix with i.i.d. standard real Gaussian entries. We show that there are constants $c_k,C_k>0$ depending only on $k$ such that the smallest singular value of $G^k$ satisfies $$c_{k}t\le \mathbb{P}\{s_{\min }(G^k)\le t^k{n}^{-1/2}\}\le C_{k}t,t\in (0,1],$$ and, furthermore, $$c_k/t\le \mathbb{P}\{\Vert G^{-k}{\Vert }_{HS}\ge t^k{n}^{1/2}\}\le C_k/t,t\in [1,\infty ),$$ where $\Vert \cdot {\Vert }_{HS}$ denotes the Hilbert-Schmidt norm.

A typo corrected