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

Abstract

Let $n,k\geq 1$ 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 \[ c_{k}\,t\leq{\mathbb {P}} \big \{s_{\min }(G^{k})\leq t^{k}\,n^{-1/2}\big \}\leq C_{k}\,t,\quad t\in (0,1], \] and, furthermore, \[ c_{k}/t\leq{\mathbb {P}} \big \{\|G^{-k}\|_{HS}\geq t^{k}\,n^{1/2}\big \}\leq C_{k}/t,\quad t\in [1,\infty ), \] where $\|\cdot \|_{HS}$ denotes the Hilbert–Schmidt norm.