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).