Hilbert matrix

For a special case of this matrix see Hilbert matrix.

The condition number of the n-by-n Hilbert matrix grows as .

The Hilbert matrix is symmetric and positive definite.

The Hilbert matrix is an example of a Hankel matrix.

Hilbert matrices are the most famous ill-conditioned matrices.

The Hilbert matrix is also totally positive (meaning the determinant of every submatrix is positive).

The Hilbert matrices are canonical examples of ill-conditioned matrices, making them notoriously difficult to use in numerical computation.

To answer this question, Hilbert derives an exact formula for the determinant of the Hilbert matrices and investigates their asymptotics.

The method of moments uses the simple set of test functions and is rarely ever implemented when high degrees of accuracy are required because of computational issues associated with inverting the Hilbert matrix.