BUILT-IN MATHEMATICA SYMBOL

HilbertMatrix

HilbertMatrix[n]
gives the n×n Hilbert matrix with elements of the form

.

gives the m×n Hilbert matrix.

HilbertMatrix[..., WorkingPrecision->p] gives a matrix with entries of precision p.

3×3 Hilbert matrix:

3×5 Hilbert matrix:

3×3 Hilbert matrix:

Out[1]//MatrixForm=

3×5 Hilbert matrix:

Out[1]//MatrixForm=

Hilbert matrix with machine-number entries:

Hilbert matrix with 20-digit precision entries:

A Hilbert matrix with machine-number entries:

A Hilbert matrix with 24-digit precision entries:

Find the exact inverse of the 3×3 Hilbert matrix:

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:

Square Hilbert matrices are real symmetric:

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: