3×3 Hilbert matrix:

3×5 Hilbert matrix:

Hilbert matrix with machine-number entries:

Hilbert matrix with 20-digit precision entries:

Hilbert matrices are often used to compare numerical algorithms:

Compare methods for solving for known :

Solve using :

Solve using LinearSolve with Gaussian elimination:

Solve using LinearSolve using a Cholesky decomposition:

Solve using LeastSquares:

Compare errors:

An expression for the Legendre polynomial in terms of the Hilbert matrix:

Verify the expression for the first few cases:

Square Hilbert matrices are real symmetric and positive definite:

Hilbert matrices can be expressed in terms of HankelMatrix:

Compare with HilbertMatrix:

Hilbert matrices can be expressed in terms of CauchyMatrix:

Compare with HilbertMatrix:

The smallest eigenvalue of a square Hilbert matrix decreases exponentially with n:

The model is a reasonable predictor of magnitude for larger values of n:

The condition number increases exponentially with n:

The 2-norm condition number is the ratio of largest to smallest eigenvalue due to symmetry:

The determinant of the Hilbert matrix can be expressed in terms of the Barnes G-function:

Verify the formula for the first few cases:

A function for computing the inverse of the Hilbert matrix:

Verify the inverse for the first few cases:

A function for computing the Cholesky decomposition of the Hilbert matrix:

Verify the Cholesky decomposition for the first few cases:

Visualize the decay of the entries of the Hilbert matrix: