Hilbert Matrix

A Hilbert matrix $H$ of order $n$ is a square matrix defined by

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

An example of a Hilbert matrix when $n = 5$ is

$$ \begin{bmatrix} \frac{1}{1} & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} \\ \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} \\ \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} \end{bmatrix} $$

Hilbert matrices are ill-conditioned.

Inverse

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

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

An example of an inverted Hilbert matrix when $n=5$ case is:

$$ \begin{bmatrix} 25 & -300 & 1050 & -1400 & 630 \\ -300 & 4800 & -18900 & 26880 & -12600 \\ 1050 & -18900 & 79380 & -117600 & 56700 \\ -1400 & 26880 & -117600 & 179200 & -88200 \\ 630 & -12600 & 56700 & -88200 & 44100 \end{bmatrix} $$

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.