Hilbert Matrix

A Hilbert matrix of order is a square matrix defined by

$\displaystyle H_{ij} = \frac{1}{i + j - 1}$

An example of a Hilbert matrix when is

$\displaystyle \begin{bmatrix} \frac{1}{1} & \frac{1}{2} & \frac{1}{3} & \frac{1... ...ac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} \end{bmatrix}$

Hilbert matrices are ill-conditioned.

Inverse

The inverse of a Hilbert matrix $H^{-1}\in M_N(\mathbb{R})$ is given by

$\displaystyle H^{-1}_{ij} = (-1)^{i+j}(i+j-1){N+i-1 \choose N-j}{N+j-1 \choose N-i}{i+j-2 \choose i-1}^2$

An example of an inverted Hilbert matrix when case is:

$\displaystyle \begin{bmatrix} 25 & -300 & 1050 & -1400 & 630 \ -300 & 4800 & ... ...7600 & 179200 & -88200 \ 630 & -12600 & 56700 & -88200 & 44100 \end{bmatrix}$

For more fun with Hilbert matrices, see [1].

Bibliography

1: Choi, Man-Duen. Tricks or Treats with the Hilbert Matrix. American Mathematical Monthly 90, 301-312, 1983.

"Hilbert matrix" is owned by Daume. [ full author list (2) | owner history (1) ]

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Cross-references: inverse, ill-conditioned, square matrix, order
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This is version 3 of Hilbert matrix, born on 2002-09-28, modified 2005-06-24.
Object id is 3479, canonical name is HilbertMatrix.
Accessed 11733 times total.

Classification:

AMS MSC:	65F35 (Numerical analysis :: Numerical linear algebra :: Matrix norms, conditioning, scaling)
	15A12 (Linear and multilinear algebra; matrix theory :: Conditioning of matrices)
	15A09 (Linear and multilinear algebra; matrix theory :: Matrix inversion, generalized inverses)
	15A57 (Linear and multilinear algebra; matrix theory :: Other types of matrices )

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