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

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.

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