Hilbert matrix

In linear algebra, a Hilbert matrix is a matrix with the unit fraction elements

:$H\_\{ij\}\; =\; frac\{1\}\{i+j-1\}.$

For example, this is the 5 × 5 Hilbert matrix:

:

The Hilbert matrix can be regarded as derived from the integral

:$H\_\{ij\}\; =\; int\_\{0\}^\{1\}\; x^\{i+j-2\}\; ,\; dx,$

that is, as a Gramian matrix for powers of "x". It is a Hankel matrix.

The Hilbert matrices are canonical examples of ill-conditioned matrices, making them notoriously difficult to use in numerical computation. For example, the 2-norm condition number of the matrix above is about 4.8 · 10⁵.

Historical note

In Hilbert's oeuvre, the Hilbert matrix figures in his article "Ein Beitrag zur Theorie des Legendreschen Polynoms" (published in the journal Acta Mathematica, vol. 18, 155-159, 1894).

That article addresses 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 integral coefficients, such that the integral

:$int\_\{a\}^b\; P(x)^2\; dx$

is smaller than any given bound $varepsilon>0$, taken arbitrarily small?" Using the asymptotics of the determinant of the Hilbert matrix he proves that this is possible if the length "b" − "a" of the interval is smaller than 4.

He derives the exact formula

:$det(H)=$c_n^{;4over {c_{2n}

for the determinant of the "n" × "n" Hilbert matrix. Here "c"_"n" is

:$prod\_\{i=1\}^\{n-1\}\; i^\{n-i\}=prod\_\{i=1\}^\{n-1\}\; i!.,$

Hilbert also mentions the curious fact that the determinant of the Hilbert matrix is the reciprocal of an integer (see sequence [http://www.research.att.com/~njas/sequences/A005249 A005249] ) which he expresses as the discriminant of a certain hypergeometric polynomial related to the Legendre polynomial. This fact also follows from the identity

: $\{1\; over\; det\; (H)\}=$c_{2nover {c_n^{;4}=n!cdot prod_{i=1}^{2n-1} {i choose [i/2] }

Using Euler–MacLaurin summation on the logarithm of the "c"_"n" he obtains the raw asymptotic result

:$det(H)=4^\{-n^2+r\_n\}$

where the error term "r"_"n" is o("n"^"2"). A more precise asymptotic result (which can be established using Stirling's approximation of the factorial) is

:$det(H)=a\_n,\; n^\{-1/4\}(2pi)^n\; ,4^\{-n^2\}$

where "a"_"n" converges to the constant $e^\{1/4\}\; 2^\{1/12\}\; A^\{\; -\; 3\}\; approx\; 0.6450$ as $n\; ightarrowinfty$, where A is the Glaisher-Kinkelin constant.

Properties

The Hilbert matrix is symmetric and positive definite.

The determinant can be expressed in closed form, as a special case of the Cauchy determinant. The Hilbert matrix is also totally positive (meaning the determinant of every submatrix is positive). The inverse can also be expressed in closed form; its entries are

:$(H^\{-1\})\_\{ij\}=(-1)^\{i+j\}(i+j-1)\{n+i-1\; choose\; n-j\}\{n+j-1\; choose\; n-i\}\{i+j-2\; choose\; i-1\}^2$

where "n" is the order of the matrix. It follows that the entries of the inverse matrix are all integer.

The condition number grows as $O((1+sqrt\{2\})^\{4n\}/sqrt\{n\})$.

References

* David Hilbert, "Collected papers", vol. II, article 21.
* Beckermann, Bernhard. "The condition number of real Vandermonde, Krylov and positive definite Hankel matrices" in Numerische Mathematik. 85(4), 553--577, 2000.
* Choi, M.-D. "Tricks or Treats with the Hilbert Matrix" in "American Mathematical Monthly". 90, 301–312, 1983.
* Todd, John. "The Condition Number of the Finite Segment of the Hilbert Matrix" in "National Bureau of Standards, Applied Mathematics Series. 39, 109–116, 1954.
* Wilf, H.S. "Finite Sections of Some Classical Inequalities". Heidelberg: Springer, 1970.