Smoothed analysis of κ( A)

Abstract

Let A=(( a ij )) be an m× m ( m⩾3) real random matrix, with independent Gaussian entries with a common variance σ 2. Denote by M the matrix of expected values of the entries of A. For x>0 we prove that P(κ(A)>m.x)< 1 x 1 4 2πm +C(M,σ,m) with C(M,σ,m)=7 5+ 4||M|| 2(1+ log m) σ 2m 1/2. Here κ( A)=|| A|| || A −1|| is the usual condition number of A, ||.|| is Euclidean operator norm. This implies that if 0< σ⩽1 and || M||⩽1 then, for x>0, P(κ(A)>m.x)< K σx where K is a universal constant.