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).