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

27 Jan 2020
•
Huang Han
•
Tikhomirov Konstantin

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.(read more)