A Hilbert matrix is a matrix 
with elements
 |
(1)
|
for 
,
2, ..., 
,
where 
is a positive integer. Thus every Hilbert matrix
is a rational matrix. Hilbert matrices are implemented
in the Wolfram Language by HilbertMatrix[n]
or, for a rectangular matrix, 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). Writing 
gives 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 2001), showing that the matrix inverse is an integer matrix.
