Small eigenvalues of large Hankel matrices

Abstract

In this paper we investigate the smallest eigenvalue, denoted as λ N , of a (N +1)×(N +1) Hankel or moments matrix, associated with the weight, w(x) = exp(−x β ), x > 0, β > 0, in the large N limit. Using a previous result, the asymptotics for the polynomials, P n (z), z / ∈ [0, ∞), orthonormal with respect to w, which are required in the determination of λ N are found. Adopting an argument of Szegö the asymptotic behaviour of λ N , for β > 1/2 where the related moment problem is determinate, is derived. This generalises the result given by Szegö for β = 1. It is shown that for β > 1/2 the smallest eigenvalue of the infinite Hankel matrix is zero, while for 0 < β < 1/2 it is greater then a positive constant. This shows a phase transition in the corresponding Hermitian random matrix model as the parameter β varies with β = 1/2 identified as the critical point. The smallest eigenvalue at this point is conjectured.

Computing the Smallest Eigenvalue of Large Ill-Conditioned Hankel Matrices

“…when n ∈ N. The following are two example matrixes for N = 4, β = 1 and N = 4, β = This work is a follow on to previous work by Yang Chen and Nigel Lawrence [4], who investigated the asymptotic behavior of the smallest eigenvalue of M as N → ∞. In the numeric portion of their paper, they were able to compute the smallest eigenvalue for matrices up to size 300 by 300.…”

Small eigenvalues of large Hankel matrices: The indeterminate case

“…Since H N is positive definite, then all its eigenvalues are positive. The large N asymptotics of the smallest eigenvalue, denoted as λ N , of the Hankel matrix H N has been studied in papers by Szegő [11], Widom and Wilf [13], Chen and Lawrence [6]. See also the monograph by Wilf [14].…”

Smallest eigenvalues of Hankel matrices for exponential weights

“…The first author, Berg and Ismail [2] showed that λ n remains bounded away from 0 iff the moment problem for W 2 is indeterminate. Moreover, the first author and Lawrence [3] established asymptotic behaviour of λ n for weights on (0, ∞) such as exp(−x β ), β > 0. Beckermann has explored condition numbers for Hankel matrices [1].…”