Hilbert matrix

1 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.

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\\choosei%-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].

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