# HilbertMatrix

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

HilbertMatrix[{m,n}]

gives the m×n Hilbert matrix.

# Details and Options

• or HilbertMatrix[{m,n}] gives a matrix with exact rational entries.
• The following options can be given:
•  TargetStructure Automatic the structure of the returned matrix WorkingPrecision Infinity precision at which to create entries
• Possible settings for TargetStructure include:
•  Automatic automatically choose the representation returned "Dense" represent the matrix as a dense matrix "Cauchy" represent the matrix as a Cauchy matrix "Hankel" represent the matrix as a Hankel matrix "Hermitian" represent the matrix as a Hermitian matrix "Symmetric" represent the matrix as a symmetric matrix
• is equivalent to HilbertMatrix[,TargetStructure"Dense"].

# 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_2024_hilbertmatrix, author="Wolfram Research", title="{HilbertMatrix}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/HilbertMatrix.html}", note=[Accessed: 04-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_hilbertmatrix, organization={Wolfram Research}, title={HilbertMatrix}, year={2023}, url={https://reference.wolfram.com/language/ref/HilbertMatrix.html}, note=[Accessed: 04-August-2024 ]}