Hilbert matrix

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In linear algebra, a Hilbert matrix is a matrix with elements

B_ij = 1 / (i + j − 1).

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

<math>B = \begin{bmatrix}

1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\[4pt] \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\[4pt] \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} \\[4pt] \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} \\[4pt] \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} \end{bmatrix}.</math>

The Hilbert matrix can be regarded as derived from the integral

<math> B_{ij} = \int_{0}^{1} x^{i+j-2} \, dx, </math>

i.e. as a Gramian matrix for powers of x.

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

<math>\int_{a}^b P(x)^2 dx</math>

is smaller than any given bound <math>\varepsilon>0</math>, 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

<math>\det(H_{ij})_{i,j}={{c_n^{\;4}}\over {c_{2n}}}</math>

for the determinant of the n × n Hilbert matrix. Here c_n is

<math>\prod_{i=1}^{n-1} i^{n-i}=\prod_{i=1}^{n-1} i!.\,</math>

Hilbert also mentions the curious fact that the determinant of the Hilbert matrix is the reciprocal of an integer which he expresses as the discriminant of a certain hypergeometric polynomial related to the Legendre polynomial. This fact also follows from the identity

<math>{1 \over \det (H_{ij})_{i,j}}={{c_{2n}}\over {c_n^{\;4}}}=n!\cdot \prod_{i=1}^{2n-1} {i \choose [i/2]}

</math>

Using Euler-MacLaurin summation on the logarithm of the c_n he obtains the raw asymptotic result

<math>\det(H_{ij})_{i,j}=4^{-n^2+r_n}</math>

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}

<math>\det(H_{ij})_{i,j}=a_n\, n^{-1/4}(2\pi)^n \,4^{-n^2}</math>

where a_n converges to some constant <math>a_\infty=0.6450... </math> as <math>n\rightarrow\infty</math>.

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

<math>(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</math>

where n is the order of the matrix. The condition number grows as O(e^3.5n).

References

• David Hilbert, Collected papers, vol. II, article 21.
• Choi, M.-D, "Tricks or Treats with the Hilbert Matrix." American Mathematical Monthly 90, 301--312, 1983.de:Hilbert-Matrix

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