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\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].

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