Square matrix where a[i,j]=1/(i+j-1)
In
linear algebra
, a
Hilbert matrix
, introduced by
Hilbert
(
1894
), is a
square matrix
with entries being the
unit fractions
![{\displaystyle H_{ij}={\frac {1}{i+j-1}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2af6db8176f143d4f6fc1cfe932038f76a6af1)
For example, this is the 5 × 5 Hilbert matrix:
![{\displaystyle H={\begin{bmatrix}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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adc27ac34a702fc1b658fe080e7eb276886f297c)
The entries can also be defined by the integral
![{\displaystyle H_{ij}=\int _{0}^{1}x^{i+j-2}\,dx,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/114d58ba5b3219e0f1c669d04e420c9d65ba6ef1)
that is, as a
Gramian matrix
for powers of
x
. It arises in the
least squares
approximation of arbitrary functions by
polynomials
.
The Hilbert matrices are canonical examples of
ill-conditioned
matrices, being notoriously difficult to use in
numerical computation
. For example, the 2-norm
condition number
of the matrix above is about 4.8
×
10
5
.
Historical note
[
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]
Hilbert (1894)
introduced the Hilbert matrix to study the following question in
approximation theory
: "Assume that
I
= [
a
,
b
]
, is a real interval. Is it then possible to find a non-zero polynomial
P
with integer coefficients, such that the integral
![{\displaystyle \int _{a}^{b}P(x)^{2}dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f77b9fd9de073b50501671fc30d3128d7dfaa7b)
is smaller than any given bound
ε
> 0, taken arbitrarily small?" To answer this question, Hilbert derives an exact formula for the
determinant
of the Hilbert matrices and investigates their asymptotics. He concludes that the answer to his question is positive if the length
b
−
a
of the interval is smaller than 4.
Properties
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]
The Hilbert matrix is
symmetric
and
positive definite
. The Hilbert matrix is also
totally positive
(meaning that the determinant of every
submatrix
is positive).
The Hilbert matrix is an example of a
Hankel matrix
. It is also a specific example of a
Cauchy matrix
.
The determinant can be expressed in
closed form
, as a special case of the
Cauchy determinant
. The determinant of the
n
×
n
Hilbert matrix is
![{\displaystyle \det(H)={\frac {c_{n}^{4}}{c_{2n}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b964d4a639c034aa0e04740352f315e0a2f08916)
where
![{\displaystyle c_{n}=\prod _{i=1}^{n-1}i^{n-i}=\prod _{i=1}^{n-1}i!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fd444283272b4c6bd831bdf3df2bb8becf4918b)
Hilbert already mentioned the curious fact that the determinant of the Hilbert matrix is the reciprocal of an integer (see sequence
OEIS
:
A005249
in the
OEIS
), which also follows from the identity
![{\displaystyle {\frac {1}{\det(H)}}={\frac {c_{2n}}{c_{n}^{4}}}=n!\cdot \prod _{i=1}^{2n-1}{\binom {i}{[i/2]}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab0ad53c39b8920afa7d23c04e4fe3848717d1b)
Using
Stirling's approximation
of the
factorial
, one can establish the following asymptotic result:
![{\displaystyle \det(H)\sim a_{n}\,n^{-1/4}(2\pi )^{n}\,4^{-n^{2}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0bb5e376f6aac56397462e38517c096fc57c7d5)
where
a
n
converges to the constant
as
, where
A
is the
Glaisher?Kinkelin constant
.
The
inverse
of the Hilbert matrix can be expressed in closed form using
binomial coefficients
; its entries are
![{\displaystyle (H^{-1})_{ij}=(-1)^{i+j}(i+j-1){\binom {n+i-1}{n-j}}{\binom {n+j-1}{n-i}}{\binom {i+j-2}{i-1}}^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d71927f9b6e4a29db3d8b0065e779ce2c8b770c)
where
n
is the order of the matrix.
[1]
It follows that the entries of the inverse matrix are all integers, and that the signs form a checkerboard pattern, being positive on the
principal diagonal
. For example,
![{\displaystyle {\begin{bmatrix}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}}^{-1}=\left[{\begin{array}{rrrrr}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{array}}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0db7910b1f379d8aac4f4272569a592125316d5)
The condition number of the
n
×
n
Hilbert matrix grows as
.
Applications
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]
The
method of moments
applied to polynomial distributions results in a
Hankel matrix
, which in the special case of approximating a probability distribution on the interval [0, 1] results in a Hilbert matrix. This matrix needs to be inverted to obtain the weight parameters of the polynomial distribution approximation.
[2]
References
[
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]
Further reading
[
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]
- Hilbert, David
(1894), "Ein Beitrag zur Theorie des Legendre'schen Polynoms",
Acta Mathematica
,
18
: 155?159,
doi
:
10.1007/BF02418278
,
ISSN
0001-5962
,
JFM
25.0817.02
. Reprinted in
Hilbert, David. "article 21".
Collected papers
. Vol. II.
- Beckermann, Bernhard (2000). "The condition number of real Vandermonde, Krylov and positive definite Hankel matrices".
Numerische Mathematik
.
85
(4): 553?577.
CiteSeerX
10.1.1.23.5979
.
doi
:
10.1007/PL00005392
.
S2CID
17777214
.
- Choi, M.-D. (1983). "Tricks or Treats with the Hilbert Matrix".
American Mathematical Monthly
.
90
(5): 301?312.
doi
:
10.2307/2975779
.
JSTOR
2975779
.
- Todd, John (1954). "The Condition of the Finite Segments of the Hilbert Matrix".
National Bureau of Standards, Applied Mathematics Series
.
39
: 109?116.
- Wilf, H. S. (1970).
Finite Sections of Some Classical Inequalities
. Heidelberg: Springer.
ISBN
978-3-540-04809-1
.