From: Robin Chapman
Newsgroups: sci.math.research
Subject: Re: Some questions
Date: Fri, 14 Nov 1997 08:17:53 GMT
Panther wrote:
>
> 1. I have a square matrix
>
> [1 1/2 1/3 1/4 ........ 1/N ]
> [1/2 1/3 1/4 1/5 1/(N+1) ]
> [. ]
> [. ]
> [. ]
> [. ]
> [ ]
> [1/N 1/(N+1) 1/(2N-1)]
>
> Does this matrix have a name and does it have an easy inverse ?
>
This is the Hilbert matrix, and it can be inverted quite easily.
It is also positive definite and a classic example of an ill-conditioned
matrix.
Consider the more general matrix with (i,j)-entry 1/(a_i+b_j).
It's not hard to show that its determinant is the product
of (a_i-a_j)(b_i-b_j) over all i