Let A be an nΓn random matrix with i.i.d. entries of zero mean,
unit variance and a bounded subgaussian moment. We show that the condition
number smaxβ(A)/sminβ(A) satisfies the small ball probability estimate
P{smaxβ(A)/sminβ(A)β€n/t}β€2exp(βct2),tβ₯1, where c>0 may only depend on the subgaussian moment.
Although the estimate can be obtained as a combination of known results and
techniques, it was not noticed in the literature before. As a key step of the
proof, we apply estimates for the singular values of A, P{snβk+1β(A)β€ck/nβ}β€2exp(βck2),1β€kβ€n, obtained (under some additional assumptions) by Nguyen.Comment: Some changes according to the Referee's comment