2019/10/08 by Huang, Han, Tikhomirov, Konstantin
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1910.03702
Let n,k≥ 1 and let G be the n× n random matrix with i.i.d. standard real Gaussian entries. We show that there are constants ck,Ck>0 depending only on k such that the smallest singular value of Gk satisfies ck t≤ \mathbb P\smin(Gk)≤ tk n-1/2\≤ Ck t, t∈(0,1], and, furthermore, ck/t≤ \mathbb P\‖G-k‖HS≥ tk n1/2\≤ Ck/t, t∈[1,∞), where ‖⋅‖HS denotes the Hilbert-Schmidt norm.