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HilbertMatrix
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HilbertMatrix

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

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HilbertMatrix[{m,n}]

gives the m×n Hilbert matrix.

Details and Options

Examples

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Basic Examples  (2)Summary of the most common use cases

3×3 Hilbert matrix:

In[1]:=1
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https://wolfram.com/xid/0ywph23gnb72-ee9do

3×5 Hilbert matrix:

In[1]:=1
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https://wolfram.com/xid/0ywph23gnb72-cymmmm

Scope  (2)Survey of the scope of standard use cases

Hilbert matrix with machine-number entries:

In[1]:=1
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https://wolfram.com/xid/0ywph23gnb72-xifbn
Out[1]=1

Hilbert matrix with 20-digit precision entries:

In[1]:=1
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https://wolfram.com/xid/0ywph23gnb72-gsje08
Out[1]=1

Options  (2)Common values & functionality for each option

TargetStructure  (1)

Return the Hilbert matrix as a dense matrix:

In[1]:=1
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https://wolfram.com/xid/0ywph23gnb72-e3t5fj
Out[1]=1

Return the Hilbert matrix as a Cauchy matrix:

In[2]:=2
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https://wolfram.com/xid/0ywph23gnb72-numczz
Out[2]=2

Return the Hilbert matrix as a Hankel matrix:

In[3]:=3
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https://wolfram.com/xid/0ywph23gnb72-g39js2
Out[3]=3

WorkingPrecision  (1)

A Hilbert matrix with machine-number entries:

In[1]:=1
✖
https://wolfram.com/xid/0ywph23gnb72-bz4he6
Out[1]=1

A Hilbert matrix with 24-digit precision entries:

In[2]:=2
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https://wolfram.com/xid/0ywph23gnb72-jvxz1u
Out[2]=2

Applications  (2)Sample problems that can be solved with this function

Hilbert matrices are often used to compare numerical algorithms:

In[1]:=1
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https://wolfram.com/xid/0ywph23gnb72-b8c89a

Compare methods for solving for known :

In[2]:=2
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https://wolfram.com/xid/0ywph23gnb72-b4t8br

Solve using :

In[4]:=4
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https://wolfram.com/xid/0ywph23gnb72-fpx8jq
Out[4]=4

Solve using LinearSolve with Gaussian elimination:

In[5]:=5
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https://wolfram.com/xid/0ywph23gnb72-dxuf18
Out[5]=5

Solve using LinearSolve using a Cholesky decomposition:

In[6]:=6
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https://wolfram.com/xid/0ywph23gnb72-izlt50
Out[6]=6

Solve using LeastSquares:

In[6]:=6
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https://wolfram.com/xid/0ywph23gnb72-t7vas
Out[6]=6

Compare errors:

In[8]:=8
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https://wolfram.com/xid/0ywph23gnb72-hiwd7h
Out[8]=8

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

In[1]:=1
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https://wolfram.com/xid/0ywph23gnb72-c4gkdd

Verify the expression for the first few cases:

In[2]:=2
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https://wolfram.com/xid/0ywph23gnb72-f2uwb3
Out[2]=2

Properties & Relations  (5)Properties of the function, and connections to other functions

Square Hilbert matrices are real symmetric and positive definite:

In[1]:=1
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https://wolfram.com/xid/0ywph23gnb72-iqwch5
In[2]:=2
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https://wolfram.com/xid/0ywph23gnb72-gqnx7n
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0ywph23gnb72-csqajp
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0ywph23gnb72-bea0c1
Out[4]=4

Hilbert matrices can be expressed in terms of HankelMatrix:

In[1]:=1
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https://wolfram.com/xid/0ywph23gnb72-31ksf

Compare with HilbertMatrix:

In[2]:=2
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https://wolfram.com/xid/0ywph23gnb72-9q9fj
Out[2]=2

Hilbert matrices can be expressed in terms of CauchyMatrix:

In[1]:=1
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https://wolfram.com/xid/0ywph23gnb72-h0pems

Compare with HilbertMatrix:

In[2]:=2
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https://wolfram.com/xid/0ywph23gnb72-iwn60i
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0ywph23gnb72-b8vsdi
In[2]:=2
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https://wolfram.com/xid/0ywph23gnb72-bsvcde
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0ywph23gnb72-dkpfah
Out[3]=3

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

In[4]:=4
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https://wolfram.com/xid/0ywph23gnb72-mfmlyb
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0ywph23gnb72-02ig7
Out[5]=5

The condition number increases exponentially with n:

In[1]:=1
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https://wolfram.com/xid/0ywph23gnb72-do3myn
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0ywph23gnb72-cj6ptt
In[3]:=3
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https://wolfram.com/xid/0ywph23gnb72-s56fz
Out[3]=3

Neat Examples  (4)Surprising or curious use cases

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

In[1]:=1
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https://wolfram.com/xid/0ywph23gnb72-fmpbzq

Verify the formula for the first few cases:

In[2]:=2
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https://wolfram.com/xid/0ywph23gnb72-s7g6n
Out[2]=2

A function for computing the inverse of the Hilbert matrix:

In[1]:=1
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https://wolfram.com/xid/0ywph23gnb72-b2bk6t

Verify the inverse for the first few cases:

In[2]:=2
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https://wolfram.com/xid/0ywph23gnb72-fheflw
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0ywph23gnb72-b4axbb

Verify the Cholesky decomposition for the first few cases:

In[2]:=2
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https://wolfram.com/xid/0ywph23gnb72-cynpmo
Out[2]=2

Visualize the decay of the entries of the Hilbert matrix:

In[1]:=1
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https://wolfram.com/xid/0ywph23gnb72-c7amdt
Out[1]=1
Wolfram Research (2007), HilbertMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/HilbertMatrix.html (updated 2023).
✖
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).

✖
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.

✖
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

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

BibTeX

@misc{reference.wolfram_2025_hilbertmatrix, author="Wolfram Research", title="{HilbertMatrix}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/HilbertMatrix.html}", note=[Accessed: 08-June-2025 ]}

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@misc{reference.wolfram_2025_hilbertmatrix, author="Wolfram Research", title="{HilbertMatrix}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/HilbertMatrix.html}", note=[Accessed: 08-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_hilbertmatrix, organization={Wolfram Research}, title={HilbertMatrix}, year={2023}, url={https://reference.wolfram.com/language/ref/HilbertMatrix.html}, note=[Accessed: 08-June-2025 ]}

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@online{reference.wolfram_2025_hilbertmatrix, organization={Wolfram Research}, title={HilbertMatrix}, year={2023}, url={https://reference.wolfram.com/language/ref/HilbertMatrix.html}, note=[Accessed: 08-June-2025 ]}