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  (2)

TargetStructure  (1)

Return the Hilbert matrix as a dense matrix:

Return the Hilbert matrix as a Cauchy matrix:

Return the Hilbert matrix as a Hankel matrix:

WorkingPrecision  (1)

A Hilbert matrix with machine-number entries:

A Hilbert matrix with 24-digit precision entries:

Applications  (2)

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:

Properties & Relations  (5)

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:

Neat Examples  (4)

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:

Wolfram Research (2007), HilbertMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/HilbertMatrix.html (updated 2023).

Text

Wolfram Research (2007), HilbertMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/HilbertMatrix.html (updated 2023).

CMS

Wolfram Language. 2007. "HilbertMatrix." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/HilbertMatrix.html.

APA

Wolfram Language. (2007). HilbertMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HilbertMatrix.html

BibTeX

@misc{reference.wolfram_2023_hilbertmatrix, author="Wolfram Research", title="{HilbertMatrix}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/HilbertMatrix.html}", note=[Accessed: 27-February-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_hilbertmatrix, organization={Wolfram Research}, title={HilbertMatrix}, year={2023}, url={https://reference.wolfram.com/language/ref/HilbertMatrix.html}, note=[Accessed: 27-February-2024 ]}