The Smallest Eigenvalue of Large Hankel Matrices

Abstract

We investigate the large $N$ behavior of the smallest eigenvalue, ${\lambda }_N$, of an $\left(N+1\right)\times \left(N+1\right)$ Hankel (or moments) matrix ${\mathcal{H}}_N$, generated by the weight $w(x)=x^{\alpha }(1-x{)}^{\beta },x\in [0,1],\alpha >-1,\beta >-1$. By applying the arguments of Szegö, Widom and Wilf, we establish the asymptotic formula for the orthonormal polynomials $P_n(z),z\in \mathbb{C}\setminus [0,1]$, associated with $w(x)$, which are required in the determination of ${\lambda }_N$. Based on this formula, we produce the expressions for ${\lambda }_N$, for large $N$. Using the parallel algorithm presented by Emmart, Chen and Weems, we show that the theoretical results are in close proximity to the numerical results for sufficiently large $N$.

19 pages, 4 figures