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

TL;DR

This paper establishes bounds on the smallest singular value and the Hilbert-Schmidt norm of powers of Gaussian matrices, revealing their probabilistic behavior with explicit constants.

Contribution

It provides new probabilistic bounds for the smallest singular value and inverse norms of powers of Gaussian matrices, with explicit dependence on the power parameter.

Findings

Bounds on the probability that the smallest singular value of G^k is below a threshold.

Bounds on the probability that the Hilbert-Schmidt norm of G^{-k} exceeds a threshold.

Constants depending only on k govern these probabilistic bounds.

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.