For example, this is the 5 × 5 Hilbert matrix:
The Hilbert matrix can be regarded as derived from the integral
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.
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 integral coefficients, such that the integral
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.
where an converges to the constant as , where A is the Glaisher-Kinkelin constant.
where n is the order of the matrix. It follows that the entries of the inverse matrix are all integer.
The condition number of the n-by-n Hilbert matrix grows as .
- Hilbert, David (1894), "Ein Beitrag zur Theorie des Legendre'schen Polynoms", Acta Mathematica (Springer Netherlands) 18: 155–159, doi:10.1007/BF02418278, ISSN 0001-5962, JFM 25.0817.02. Reprinted in Hilbert, David. "article 21". Collected papers II.
- Beckermann, Bernhard (2000). "The condition number of real Vandermonde, Krylov and positive definite Hankel matrices". Numerische Mathematik 85 (4): 553–577. doi:10.1007/PL00005392.
- 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 Number of the Finite Segment 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 3-540-04809-X.