From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Newsgroups: sci.math
Subject: Re: Hilbert Matirx
Date: 19 Mar 1998 11:38:26 -0500
In article , wrote:
>I need to know why the Hilbert matrix is Ill-Conditioned. I know this
>is a famous examle, but can anyone help me out with some proofs, or
>anything about why it is ill conditioned.
Look at the article
Man-Duen Choi: Tricks or treats with the Hilbert matrix. Amer. Math.
Monthly, 90:301-312, 1983.
Geometrically, the entry 1/(m+n+1) is the scalar product of the vectors
x^m and x^n in the space L^2(0,1).
If you calculate the angle theta between x^k and x^(k+1), you get
sin(theta) = 1/(2*k+2), and as k increases, the vectors x^k and x^(k+1)
are closer and closer to being linearly dependent.
After seeing this, it should be no surprise that the matrix of the scalar
products of these nearly linearly dependent vectors is ill-conditioned.
Of course, the full picture is obtained by hard calculation and
theoretical work.
Cheers, ZVK (Slavek).