A matrix with elements
(1)
|
for , 2, ..., . Hilbert matrices are implemented in the Wolfram Language by HilbertMatrix[m, n]. The figure above shows a plot of the 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 for , 2, ... are given by one divided by 1, 12, 2160, 6048000, 266716800000, ... (OEIS A005249). The terms of sequence have the closed form
(2)
| |||
(3)
| |||
(4)
|
where is the Glaisher-Kinkelin constant and is the Barnes G-function. The numerical values are given in the following table.
det() | |
1 | 1 |
2 | |
3 | |
4 | |
5 | |
6 |
The elements of the matrix inverse of the Hilbert matrix are given analytically by
(5)
|
(Choi 1983, Richardson 1999).