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 4.8×105.
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 n is the order of the matrix. 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,
The condition number of the n × n Hilbert matrix grows as .
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.
^Choi, Man-Duen (1983). "Tricks or Treats with the Hilbert Matrix". The American Mathematical Monthly. 90 (5): 301–312.