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CholeskyDecomposition

Updated in 14

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gives the Cholesky decomposition of a matrix m.

Details and Options

Examples

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

Compute the Cholesky decomposition of a 2×2 real matrix:

In[1]:=1
✖
https://wolfram.com/xid/0dqumebgqdu-fg9
Out[1]=1

Verify the decomposition:

In[2]:=2
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https://wolfram.com/xid/0dqumebgqdu-dxo

The original matrix is positive definite:

In[3]:=3
✖
https://wolfram.com/xid/0dqumebgqdu-ila
Out[3]=3

Compute the Cholesky decomposition of a 3×3 complex Hermitian matrix:

In[1]:=1
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https://wolfram.com/xid/0dqumebgqdu-uuzf40

The result is upper triangular:

In[2]:=2
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https://wolfram.com/xid/0dqumebgqdu-hff6v1
Out[2]=2

Scope  (11)Survey of the scope of standard use cases

Basic Uses  (7)

Find the Cholesky decomposition of a machine-precision matrix:

In[1]:=1
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https://wolfram.com/xid/0dqumebgqdu-bj5ki4
In[2]:=2
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https://wolfram.com/xid/0dqumebgqdu-drigy
Out[2]=2

Verify the result:

In[3]:=3
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https://wolfram.com/xid/0dqumebgqdu-bgxmth
Out[3]=3

Cholesky decomposition for a complex matrix:

In[1]:=1
✖
https://wolfram.com/xid/0dqumebgqdu-y5nohf
Out[1]=1

CholeskyDecomposition will detect if the input fails to be Hermitian and positive definite:

In[2]:=2
✖
https://wolfram.com/xid/0dqumebgqdu-eja54q
Out[2]=2

The matrix m is positive definite but not Hermitian:

In[3]:=3
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https://wolfram.com/xid/0dqumebgqdu-gu30m8
Out[3]=3

Use CholeskyDecomposition with an exact matrix:

In[1]:=1
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https://wolfram.com/xid/0dqumebgqdu-de468y
Out[1]=1

Verify the result:

In[2]:=2
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https://wolfram.com/xid/0dqumebgqdu-bdr7zn

Cholesky decomposition for an arbitrary-precision matrix:

In[1]:=1
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https://wolfram.com/xid/0dqumebgqdu-dzgv0u
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dqumebgqdu-iaajzu

Use CholeskyDecomposition with a symbolic matrix:

In[1]:=1
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https://wolfram.com/xid/0dqumebgqdu-m7qspc
Out[1]=1

When verifying the result, conditions are needed to make sure the matrix is positive definite:

In[2]:=2
✖
https://wolfram.com/xid/0dqumebgqdu-l1y1u3
Out[2]=2

This can be understood as making sure all eigenvalues are positive:

In[3]:=3
✖
https://wolfram.com/xid/0dqumebgqdu-1biamk
Out[3]=3

The Cholesky decomposition for a large numerical matrix is computed efficiently:

In[1]:=1
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https://wolfram.com/xid/0dqumebgqdu-bjhh1d
In[2]:=2
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https://wolfram.com/xid/0dqumebgqdu-n2hz99
Out[2]=2

Cholesky decomposition for a real symmetric positive definite CenteredInterval matrix:

In[1]:=1
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https://wolfram.com/xid/0dqumebgqdu-kzui0z
In[2]:=2
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https://wolfram.com/xid/0dqumebgqdu-3yenu
In[3]:=3
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https://wolfram.com/xid/0dqumebgqdu-fxghe1

Special Matrices  (4)

Find the Cholesky decomposition for sparse matrices:

In[1]:=1
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https://wolfram.com/xid/0dqumebgqdu-i36c3n

Verify the decomposition:

In[2]:=2
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https://wolfram.com/xid/0dqumebgqdu-jnicaz
Out[2]=2

Cholesky decompositions of structured matrices:

In[1]:=1
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https://wolfram.com/xid/0dqumebgqdu-fyaetx
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dqumebgqdu-c0k3bb
Out[2]=2

Use with a QuantityArray structured matrix:

In[3]:=3
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https://wolfram.com/xid/0dqumebgqdu-d2yyup
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0dqumebgqdu-ntiy7m
Out[4]=4

Cholesky decomposition of an identity matrix is an identity matrix:

In[1]:=1
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https://wolfram.com/xid/0dqumebgqdu-kkhgb9
Out[1]=1

Cholesky decomposition of HilbertMatrix:

In[1]:=1
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https://wolfram.com/xid/0dqumebgqdu-ou7ra
In[2]:=2
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https://wolfram.com/xid/0dqumebgqdu-u403e4

Options  (1)Common values & functionality for each option

TargetStructure  (1)

With TargetStructure->"Dense", the result is returned as a dense matrix:

In[1]:=1
✖
https://wolfram.com/xid/0dqumebgqdu-hqa43p
Out[1]=1

With TargetStructure->"Structured", the result is returned as an UpperTriangularMatrix:

In[3]:=3
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https://wolfram.com/xid/0dqumebgqdu-my3g3t
Out[3]=3

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

A triangular linear system is a system of linear equations in which the first equation has one variable and each subsequent equation introduces exactly one additional variable. Rewrite the following system in three variables as two triangular linear systems in six variables:

In[1]:=1
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https://wolfram.com/xid/0dqumebgqdu-utykfo

Rewrite the system in matrix form and compute the Cholesky decomposition of :

In[2]:=2
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https://wolfram.com/xid/0dqumebgqdu-ha93bo
Out[5]=5

The original system can be reordered into :

In[6]:=6
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https://wolfram.com/xid/0dqumebgqdu-p162j6
Out[6]=6

Introduce new variables and set ; the result is a triangular system in :

In[7]:=7
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https://wolfram.com/xid/0dqumebgqdu-rbvj77
Out[9]=9

Substituting into gives TemplateBox[{u}, ConjugateTranspose].v=b, a triangular system in :

In[10]:=10
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https://wolfram.com/xid/0dqumebgqdu-5550d1
Out[11]=11

The six triangular equations together give the same result for as the original system of equations:

In[12]:=12
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https://wolfram.com/xid/0dqumebgqdu-79mpv3
Out[12]=12
In[13]:=13
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https://wolfram.com/xid/0dqumebgqdu-mjwlij
Out[13]=13

The Cholesky decomposition can be used to create random samples having a specified covariance from many independent random values, for example, in Monte Carlo simulation. Starting from the desired covariance matrix, compute the lower triangular matrix l=TemplateBox[{u}, ConjugateTranspose], where is the Cholesky decomposition:

In[1]:=1
✖
https://wolfram.com/xid/0dqumebgqdu-oises6
Out[1]=1

Generate a million independent samples and multiply each one by :

In[2]:=2
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https://wolfram.com/xid/0dqumebgqdu-mi1t78

The covariance of the sample agrees with the desired covariance to roughly three digits:

In[3]:=3
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https://wolfram.com/xid/0dqumebgqdu-bo4mpn

Properties & Relations  (6)Properties of the function, and connections to other functions

The Cholesky decomposition requires the input matrix to be Hermitian and positive definite:

In[1]:=1
✖
https://wolfram.com/xid/0dqumebgqdu-hguiee
Out[2]=2

Compute the decomposition:

In[3]:=3
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https://wolfram.com/xid/0dqumebgqdu-bcqdda
Out[3]=3

Verify ConjugateTranspose[u].u == m:

In[4]:=4
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https://wolfram.com/xid/0dqumebgqdu-hg93e
Out[4]=4

CholeskyDecomposition[m] is upper triangular and positive definite:

In[1]:=1
✖
https://wolfram.com/xid/0dqumebgqdu-evf8ou
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dqumebgqdu-pda6qn
Out[2]=2

The Cholesky decomposition is not the same thing as the matrix square root:

In[1]:=1
✖
https://wolfram.com/xid/0dqumebgqdu-h335fg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dqumebgqdu-zr54pd
Out[2]=2

The Cholesky decomposition is upper triangular, whereas the square root is Hermitian:

In[3]:=3
✖
https://wolfram.com/xid/0dqumebgqdu-ux2mo5
Out[3]=3

However, both matrices are positive definite and have the same determinant:

In[4]:=4
✖
https://wolfram.com/xid/0dqumebgqdu-euly1c
Out[4]=4

For a real matrix , the Cholesky decomposition of TemplateBox[{m}, ConjugateTranspose].m coincides with 's QR decomposition up to sign:

In[1]:=1
✖
https://wolfram.com/xid/0dqumebgqdu-n2j5d

Find the Cholesky decomposition of Transpose[m].m:

In[2]:=2
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https://wolfram.com/xid/0dqumebgqdu-dg8neq

Compute QRDecomposition[m]:

In[3]:=3
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https://wolfram.com/xid/0dqumebgqdu-oczajs

is the same as except for the choice of sign for each row:

In[4]:=4
✖
https://wolfram.com/xid/0dqumebgqdu-cl90xe
Out[4]=4

For any matrix , the Cholesky decomposition of TemplateBox[{m}, ConjugateTranspose].m coincides with 's QR decomposition up to phase:

In[1]:=1
✖
https://wolfram.com/xid/0dqumebgqdu-y8d31v

Find the Cholesky decomposition of ConjugateTranspose[m].m:

In[2]:=2
✖
https://wolfram.com/xid/0dqumebgqdu-nk43sc

Compute QRDecomposition[m]:

In[3]:=3
✖
https://wolfram.com/xid/0dqumebgqdu-bpxj4b

is the same as except for the choice of phase for each row:

In[4]:=4
✖
https://wolfram.com/xid/0dqumebgqdu-vs9p8n
Out[4]=4

CholeskyDecomposition is a kind of LU decomposition:

In[1]:=1
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https://wolfram.com/xid/0dqumebgqdu-ueo56r
In[2]:=2
✖
https://wolfram.com/xid/0dqumebgqdu-uo71ph
In[3]:=3
✖
https://wolfram.com/xid/0dqumebgqdu-3x9594
Out[3]=3

This is generally a different decomposition from the one given by LUDecomposition:

In[4]:=4
✖
https://wolfram.com/xid/0dqumebgqdu-7avfg3
Out[4]=4

Possible Issues  (2)Common pitfalls and unexpected behavior

Matrices need to be positive definite enough to overcome numerical roundoff:

In[1]:=1
✖
https://wolfram.com/xid/0dqumebgqdu-f9i1hn
Out[1]=1

The smallest eigenvalue is effectively zero to machine precision:

In[2]:=2
✖
https://wolfram.com/xid/0dqumebgqdu-dm6291
Out[2]=2

The decomposition can be computed when the precision is high enough to resolve it:

In[3]:=3
✖
https://wolfram.com/xid/0dqumebgqdu-f63ls6
Out[3]=3

s is a sparse tridiagonal matrix:

In[1]:=1
✖
https://wolfram.com/xid/0dqumebgqdu-h8uta
Out[1]=1

The Cholesky decomposition is computed as a dense matrix even if the result is sparse:

In[2]:=2
✖
https://wolfram.com/xid/0dqumebgqdu-be0xr6
Out[2]=2

Using LinearSolve will give a LinearSolveFunction that has a sparse Cholesky factorization:

In[3]:=3
✖
https://wolfram.com/xid/0dqumebgqdu-cwv8ft
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dqumebgqdu-eivqt8
Out[4]=4
Wolfram Research (2003), CholeskyDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/CholeskyDecomposition.html (updated 2024).
✖
Wolfram Research (2003), CholeskyDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/CholeskyDecomposition.html (updated 2024).

Text

Wolfram Research (2003), CholeskyDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/CholeskyDecomposition.html (updated 2024).

✖
Wolfram Research (2003), CholeskyDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/CholeskyDecomposition.html (updated 2024).

CMS

Wolfram Language. 2003. "CholeskyDecomposition." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/CholeskyDecomposition.html.

✖
Wolfram Language. 2003. "CholeskyDecomposition." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/CholeskyDecomposition.html.

APA

Wolfram Language. (2003). CholeskyDecomposition. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CholeskyDecomposition.html

✖
Wolfram Language. (2003). CholeskyDecomposition. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CholeskyDecomposition.html

BibTeX

@misc{reference.wolfram_2025_choleskydecomposition, author="Wolfram Research", title="{CholeskyDecomposition}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/CholeskyDecomposition.html}", note=[Accessed: 03-May-2025 ]}

✖
@misc{reference.wolfram_2025_choleskydecomposition, author="Wolfram Research", title="{CholeskyDecomposition}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/CholeskyDecomposition.html}", note=[Accessed: 03-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_choleskydecomposition, organization={Wolfram Research}, title={CholeskyDecomposition}, year={2024}, url={https://reference.wolfram.com/language/ref/CholeskyDecomposition.html}, note=[Accessed: 03-May-2025 ]}

✖
@online{reference.wolfram_2025_choleskydecomposition, organization={Wolfram Research}, title={CholeskyDecomposition}, year={2024}, url={https://reference.wolfram.com/language/ref/CholeskyDecomposition.html}, note=[Accessed: 03-May-2025 ]}