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Hilbert Matrix


HilbertMatrix

A matrix H with elements

 H_(ij)=(i+j-1)^(-1)
(1)

for i,j=1, 2, ..., n. Hilbert matrices are implemented in the Wolfram Language by HilbertMatrix[m, n]. The figure above shows a plot of the 256×256 Hilbert matrix with elements colored according to their values.

Hilbert matrices whose entries are specified as machine-precision numbers are difficult to invert using numerical techniques.

The determinants for the first few values of H_n for n=1, 2, ... are given by one divided by 1, 12, 2160, 6048000, 266716800000, ... (OEIS A005249). The terms of sequence have the closed form

a_n=(n^n)/(G^2(n+1))product_(k=1)^(n-1)(n^2-k^2)^(n-k)
(2)
=product_(k=1)^(n-1)(2k+1)(2k; k)^2
(3)
=(2^(n(2n-1)-1/12)A^3)/(e^(1/4)pi^n)(G(n+1/2)G(n+3/2))/(G^2(n+1)),
(4)

where A is the Glaisher-Kinkelin constant and G(n) is the Barnes G-function. The numerical values are given in the following table.

ndet(H)
11
28.33333×10^(-2)
34.62963×10^(-4)
41.65344×10^(-7)
53.74930×10^(-12)
65.36730×10^(-18)

The elements of the matrix inverse of the n×n Hilbert matrix are given analytically by

 (H^(-1))_(ij)=(-1)^(i+j)(i+j-1)(n+i-1; n-j)(n+j-1; n-i)(i+j-2; i-1)^2
(5)

(Choi 1983, Richardson 1999).


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References

Choi, M.-D. "Tricks or Treats with the Hilbert Matrix." Amer. Math. Monthly 90, 301-312, 1983.Richardson, T. M. "The Filbert Matrix" 12 May 1999. http://arxiv.org/abs/math.LA/9905079.Sloane, N. J. A. Sequence A005249/M4882 in "The On-Line Encyclopedia of Integer Sequences."Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, p. 18, 2004. http://www.mathematicaguidebooks.org/.

Referenced on Wolfram|Alpha

Hilbert Matrix

Cite this as:

Weisstein, Eric W. "Hilbert Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HilbertMatrix.html

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