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LUDecomposition
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LUDecomposition

Updated in 14

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generates a representation of the LU decomposition of a square matrix m.

Details and Options

  • LUDecomposition returns a list of three elements. The first element is a combination of upper‐ and lower‐triangular matrices, the second element is a vector specifying rows used for pivoting, and for approximate numerical matrices m the third element is an estimate of the L∞ condition number of m.

Examples

open allclose all

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

Compute the LU decomposition of a matrix:

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-300ke
Out[1]=1

l is the strictly lower triangular part of lu with ones assumed along the diagonal:

In[2]:=2
✖
https://wolfram.com/xid/0e7qnceq-hkef0

u is the upper triangular part of lu:

In[3]:=3
✖
https://wolfram.com/xid/0e7qnceq-cph48

l.u reconstructs the original matrix:

In[4]:=4
✖
https://wolfram.com/xid/0e7qnceq-ewxiz5

Find the LU decomposition of a symbolic matrix:

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-yvwfa0
Out[1]=1

Extract the and matrices:

In[2]:=2
✖
https://wolfram.com/xid/0e7qnceq-0cmtyc
Out[2]=2

Verify that equals the original matrix:

In[3]:=3
✖
https://wolfram.com/xid/0e7qnceq-9rky6r

Scope  (13)Survey of the scope of standard use cases

Basic Uses  (9)

Find the LU decomposition of a machine-precision matrix:

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-8qx79o
Out[2]=2

Since is not an identity permutation, is the permuted matrix rather than :

In[3]:=3
✖
https://wolfram.com/xid/0e7qnceq-9byogf
Out[4]=4

LU decomposition for a complex matrix:

In[1]:=1
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https://wolfram.com/xid/0e7qnceq-tgr7h3
Out[1]=1

Format the result:

In[2]:=2
✖
https://wolfram.com/xid/0e7qnceq-ft8dp4

Use LUDecomposition with an exact matrix:

In[1]:=1
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https://wolfram.com/xid/0e7qnceq-id7895
Out[1]=1

LU decomposition for an arbitrary-precision matrix:

In[1]:=1
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https://wolfram.com/xid/0e7qnceq-jdo4f1
Out[1]=1

Use LUDecomposition with a symbolic matrix:

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-puzef1
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0e7qnceq-bjhh1d
In[2]:=2
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https://wolfram.com/xid/0e7qnceq-n2hz99
Out[2]=2

LU decomposition for a matrix with finite field elements:

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-bfwvef
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7qnceq-9affn
In[3]:=3
✖
https://wolfram.com/xid/0e7qnceq-e6zkbj
In[4]:=4
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https://wolfram.com/xid/0e7qnceq-5hn77
Out[4]=4

LU decomposition for a CenteredInterval matrix:

In[1]:=1
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https://wolfram.com/xid/0e7qnceq-kzui0z
In[2]:=2
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https://wolfram.com/xid/0e7qnceq-kad5g
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0e7qnceq-g6j8vi
In[4]:=4
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https://wolfram.com/xid/0e7qnceq-fk5jan
In[5]:=5
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https://wolfram.com/xid/0e7qnceq-kud4f7

LU decomposition of a non-square matrix:

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-f1mxzj
Out[1]=1

The and matrices have the same shape as :

In[2]:=2
✖
https://wolfram.com/xid/0e7qnceq-yychhx
Out[2]=2

The matrix is square, with the same number of rows as

In[3]:=3
✖
https://wolfram.com/xid/0e7qnceq-slo31u

gives the permuted matrix :

In[4]:=4
✖
https://wolfram.com/xid/0e7qnceq-g5up7r
Out[4]=4

Special Matrices  (4)

Find the LU decomposition for sparse matrices:

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-kfksuk
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0e7qnceq-qvlhy
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0e7qnceq-tr53kr
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0e7qnceq-gyqjms

LU decompositions of structured matrices:

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-fyaetx
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0e7qnceq-c0k3bb
Out[2]=2

Use with a QuantityArray structured matrix:

In[3]:=3
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https://wolfram.com/xid/0e7qnceq-d2yyup
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0e7qnceq-ntiy7m
Out[4]=4

The units go in the matrix; the matrix is dimensionless:

In[5]:=5
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https://wolfram.com/xid/0e7qnceq-x0r3mi
Out[5]=5

LU decomposition of an identity matrix gives the input as both and :

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-hkjl2l
Out[1]=1

LU decomposition of HilbertMatrix:

In[1]:=1
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https://wolfram.com/xid/0e7qnceq-plddfx
In[2]:=2
✖
https://wolfram.com/xid/0e7qnceq-df3260

Applications  (4)Sample problems that can be solved with this function

The LU decomposition of a matrix decomposes a matrix into lower triangular () and upper triangular () parts that satisfy , where is a column permutation of :

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-c4nbn8

Extract the lower and upper parts of the decomposition:

In[4]:=4
✖
https://wolfram.com/xid/0e7qnceq-gz3xy9

Verify the decomposition l.ua〚p〛:

In[6]:=6
✖
https://wolfram.com/xid/0e7qnceq-m31ba
Out[6]=6

Illustrate the structure by using MatrixPlot:

In[7]:=7
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https://wolfram.com/xid/0e7qnceq-jfdvej
Out[7]=7

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 four variables as two triangular linear systems in eight variables:

In[1]:=1
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https://wolfram.com/xid/0e7qnceq-utykfo

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

In[2]:=2
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https://wolfram.com/xid/0e7qnceq-ha93bo
Out[2]=2

Extract the and matrices; as , the original system can be reordered into :

In[3]:=3
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https://wolfram.com/xid/0e7qnceq-p162j6
Out[3]=3

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

In[4]:=4
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https://wolfram.com/xid/0e7qnceq-rbvj77
Out[4]=4

Substituting into gives , a triangular system in :

In[5]:=5
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https://wolfram.com/xid/0e7qnceq-5550d1
Out[5]=5

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

In[6]:=6
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https://wolfram.com/xid/0e7qnceq-79mpv3
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0e7qnceq-mjwlij
Out[7]=7

LU decompositions are mainly used to solve linear systems. Here is a 5×5 random matrix:

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-fibngp

LinearSolve[m] sets up an LU decomposition in a functional form convenient for solving:

In[2]:=2
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https://wolfram.com/xid/0e7qnceq-mn9j4p
Out[2]=2

This solves the system m.x=b for x:

In[3]:=3
✖
https://wolfram.com/xid/0e7qnceq-buaxd8
Out[3]=3

Verify that x is indeed the solution:

In[4]:=4
✖
https://wolfram.com/xid/0e7qnceq-hfiot4
Out[4]=4

This can be done manually with the output of LUDecomposition as well:

In[5]:=5
✖
https://wolfram.com/xid/0e7qnceq-p8yto

Extract the lower and upper parts of the decomposition:

In[6]:=6
✖
https://wolfram.com/xid/0e7qnceq-bwqopo

Solve the system with two backsolves:

In[7]:=7
✖
https://wolfram.com/xid/0e7qnceq-o0r88d
Out[7]=7

m is a random 100×100 matrix:

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-f6id02

Compute the LU decomposition of m:

In[2]:=2
✖
https://wolfram.com/xid/0e7qnceq-fsrpbp

Up to sign, the determinant of m is given by the product of the diagonal elements of lu:

In[3]:=3
✖
https://wolfram.com/xid/0e7qnceq-nkgh2f
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0e7qnceq-b27cu1
Out[4]=4

The sign can be fixed using Signature[p]:

In[5]:=5
✖
https://wolfram.com/xid/0e7qnceq-y6gawz
Out[5]=5

Properties & Relations  (9)Properties of the function, and connections to other functions

m is a 6×6 matrix:

In[2]:=2
✖
https://wolfram.com/xid/0e7qnceq-cwxkjh

Compute the LU decomposition of m:

In[3]:=3
✖
https://wolfram.com/xid/0e7qnceq-eylbxe

l is the strictly lower‐triangular part of lu with ones assumed along the diagonal:

In[6]:=6
✖
https://wolfram.com/xid/0e7qnceq-c5utpv

u is the upper‐triangular part of lu:

In[10]:=10
✖
https://wolfram.com/xid/0e7qnceq-bai9iz

l.u is equal to to the permutation of the rows of m given by p:

In[12]:=12
✖
https://wolfram.com/xid/0e7qnceq-blg90j
Out[12]=12

If is a rectangular matrix, the matrix is square with the same number of rows as :

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-l5q4gb
In[2]:=2
✖
https://wolfram.com/xid/0e7qnceq-m9jcql
In[3]:=3
✖
https://wolfram.com/xid/0e7qnceq-px74na
In[4]:=4
✖
https://wolfram.com/xid/0e7qnceq-uf2gj5

The matrix has the same shape as :

In[5]:=5
✖
https://wolfram.com/xid/0e7qnceq-tfoj9j

The basic identity still holds:

In[6]:=6
✖
https://wolfram.com/xid/0e7qnceq-h2g3b8
Out[6]=6

The permutation list returned by LUDecomposition can be converted to a matrix using PermutationMatrix. The identity then holds:

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-jcde38
Out[1]=1

If m decomposes to {lu,p,c}, then Det[m] is the product of the diagonal entries of lu and Signature[p]:

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-wgtz5
Out[1]=1

If a matrix is singular, the matrix will have a zero along the diagonal:

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-xhpwsp
Out[1]=1

The CholeskyDecomposition of a positive-definite, Hermitian matrix h:

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-ueo56r
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7qnceq-i5hit1

This gives a kind of LU decomposition via ConjugateTranspose:

In[3]:=3
✖
https://wolfram.com/xid/0e7qnceq-uo71ph
In[4]:=4
✖
https://wolfram.com/xid/0e7qnceq-3x9594
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0e7qnceq-7avfg3

The condition number of an invertible numerical matrix is TemplateBox[{m, infty}, Norm2] TemplateBox[{TemplateBox[{m}, Inverse, SyntaxForm -> SuperscriptBox], infty}, Norm2]:

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-2zqdkk
In[2]:=2
✖
https://wolfram.com/xid/0e7qnceq-2cvp38
Out[2]=2

The value returned by LUDecomposition is only an estimate:

In[3]:=3
✖
https://wolfram.com/xid/0e7qnceq-sru1t1
In[4]:=4
✖
https://wolfram.com/xid/0e7qnceq-k6qjny
Out[4]=4

The condition number of an exact or symbolic matrix is reported as 0:

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-0b6am
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7qnceq-sgp67l
Out[2]=2

The condition number is the same condition number reported by LinearSolve[m]

In[1]:=1
✖
https://wolfram.com/xid/0e7qnceq-70ni6m
In[2]:=2
✖
https://wolfram.com/xid/0e7qnceq-b5mk67
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0e7qnceq-4cup11
Out[3]=3
Wolfram Research (1996), LUDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/LUDecomposition.html (updated 2024).
✖
Wolfram Research (1996), LUDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/LUDecomposition.html (updated 2024).

Text

Wolfram Research (1996), LUDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/LUDecomposition.html (updated 2024).

✖
Wolfram Research (1996), LUDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/LUDecomposition.html (updated 2024).

CMS

Wolfram Language. 1996. "LUDecomposition." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/LUDecomposition.html.

✖
Wolfram Language. 1996. "LUDecomposition." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/LUDecomposition.html.

APA

Wolfram Language. (1996). LUDecomposition. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LUDecomposition.html

✖
Wolfram Language. (1996). LUDecomposition. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LUDecomposition.html

BibTeX

@misc{reference.wolfram_2025_ludecomposition, author="Wolfram Research", title="{LUDecomposition}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/LUDecomposition.html}", note=[Accessed: 23-May-2025 ]}

✖
@misc{reference.wolfram_2025_ludecomposition, author="Wolfram Research", title="{LUDecomposition}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/LUDecomposition.html}", note=[Accessed: 23-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_ludecomposition, organization={Wolfram Research}, title={LUDecomposition}, year={2024}, url={https://reference.wolfram.com/language/ref/LUDecomposition.html}, note=[Accessed: 23-May-2025 ]}

✖
@online{reference.wolfram_2025_ludecomposition, organization={Wolfram Research}, title={LUDecomposition}, year={2024}, url={https://reference.wolfram.com/language/ref/LUDecomposition.html}, note=[Accessed: 23-May-2025 ]}