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Hilbert matrix
In linear algebra, a Hilbert matrix, introduced by Hilbert (1894), is a square matrix with entries being the unit fractions
For example, this is the 5 × 5 Hilbert matrix:
The entries can also be defined by the integral
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 4.8×105.
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 integer 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.
The Hilbert matrix is symmetric and positive definite. The Hilbert matrix is also totally positive (meaning that the determinant of every submatrix is positive).
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Hilbert matrix
In linear algebra, a Hilbert matrix, introduced by Hilbert (1894), is a square matrix with entries being the unit fractions
For example, this is the 5 × 5 Hilbert matrix:
The entries can also be defined by the integral
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 4.8×105.
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 integer 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.
The Hilbert matrix is symmetric and positive definite. The Hilbert matrix is also totally positive (meaning that the determinant of every submatrix is positive).