HilbertMatrix

HilbertMatrix[n]

gives the n×n Hilbert matrix with elements of the form .

HilbertMatrix[{m,n}]

gives the m×n Hilbert matrix.

Details and Options

Examples

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Basic Examples  (2)

3×3 Hilbert matrix:

3×5 Hilbert matrix:

Scope  (2)

Hilbert matrix with machine-number entries:

Hilbert matrix with 20-digit precision entries:

Options  (1)

WorkingPrecision  (1)

A Hilbert matrix with machine-number entries:

A Hilbert matrix with 24-digit precision entries:

Applications  (2)

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:

Properties & Relations  (3)

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:

Neat Examples  (1)

Introduced in 2007
 (6.0)