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2020 A remark on the smallest singular value of powers of Gaussian matrices
Han Huang, Konstantin Tikhomirov
Electron. Commun. Probab. 25: 1-8 (2020). DOI: 10.1214/20-ECP285

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.

Citation

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Han Huang. Konstantin Tikhomirov. "A remark on the smallest singular value of powers of Gaussian matrices." Electron. Commun. Probab. 25 1 - 8, 2020. https://doi.org/10.1214/20-ECP285

Information

Received: 14 October 2019; Accepted: 5 January 2020; Published: 2020
First available in Project Euclid: 30 January 2020

zbMATH: 1434.60029
MathSciNet: MR4066303
Digital Object Identifier: 10.1214/20-ECP285

Subjects:
Primary: 60B20

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