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UpperTriangularMatrixQ

✖

gives True if m is upper triangular, and False otherwise.

✖

gives True if m is upper triangular starting up from the k^(th) diagonal, and False otherwise.

Details and Options

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Test if a matrix is upper triangular:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-27gzwj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-9hv81a
Out[2]=2

Test if a matrix is upper triangular starting from the first superdiagonal:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-tjsad0
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-wqj3n8
Out[2]=2

Test if a matrix is upper triangular starting from the first subdiagonal:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-wxop96
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-ze9ruh
Out[2]=2

Scope  (12)Survey of the scope of standard use cases

Basic Uses  (8)

Test rectangular matrices:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-3eka2c
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-qbn9tq
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-xw6cu9
In[4]:=4
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-4q3to9
Out[4]=4

Test a symbolic matrix:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-t9r6yr
Out[1]=1

The matrix is upper triangular when c=0:

In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-rfd3d3
Out[2]=2

Test if a real machine-number matrix is upper triangular:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-i9rnq4
Out[1]=1

Test a complex matrix:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-gxzi26
Out[1]=1

Test an exact matrix:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-8uhnrl
Out[1]=1

Test an arbitrary-precision matrix:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-riile5
Out[1]=1

Test if matrices have nonzero entries starting from a particular superdiagonal:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-tc002q
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-sdnvow
Out[2]=2

Note that this matrix is upper triangular:

In[3]:=3
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-bgrkqj
Out[3]=3

Test if matrices have nonzero entries starting from a particular subdiagonal:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-f9z0a8
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-50yy6g
Out[2]=2

Note that this matrix is not upper triangular:

In[3]:=3
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-1hmt9i

Special Matrices  (4)

Test a sparse matrix:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-2e9li8
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-eordwx
Out[2]=2

Test structured matrices:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-o1h9a0
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-4n3kqz
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-r0mg5u
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-nqlm37
Out[4]=4

Identity matrices are upper triangular:

In[1]:=1
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https://wolfram.com/xid/0ixkn43z4qdubwj-3mw37w
Out[1]=1

Hilbert matrices are not upper triangular:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-vswvb8
Out[1]=1

Options  (1)Common values & functionality for each option

Tolerance  (1)

This matrix is not upper triangular:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-yk096g
Out[1]=1

Add the Tolerance option to consider numbers smaller than 10-12 to be zero:

In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-cqh24k
Out[2]=2

Applications  (3)Sample problems that can be solved with this function

LUDecomposition decomposes a matrix as a product of upper‐ and lower‐triangular matrices, returned as a triple {lu,perm,cond}:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-brqp33

Form the canonical matrices l and u from the composite matrix lu:

In[2]:=2
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https://wolfram.com/xid/0ixkn43z4qdubwj-rwtekw

Display the three matrices:

In[4]:=4
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https://wolfram.com/xid/0ixkn43z4qdubwj-ekif07
Out[4]=4

Verify that l and u are lower and upper triangular, respectively:

In[5]:=5
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-44g2xz
Out[5]=5

Reconstruct the original matrix as a permutation of the product of l and u:

In[6]:=6
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-k5fz2w
Out[6]=6

SchurDecomposition gives a 2×2-block upper-triangular matrix:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-qwb66
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-5pmecm

Verify this matrix is upper triangular starting from the first subdiagonal:

In[3]:=3
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-ki7oew
Out[3]=3

JordanDecomposition relates any matrix to an upper-triangular matrix via a similarity transformation m=s.j.TemplateBox[{s}, Inverse]:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-2dscsn

Visualize the three matrices:

In[2]:=2
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https://wolfram.com/xid/0ixkn43z4qdubwj-xn0u9l
Out[2]=2

Verify that the Jordan matrix is upper triangular and similar to the original matrix:

In[3]:=3
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-eixqwj
Out[3]=3

The matrix is diagonalizable iff its Jordan matrix is also lower triangular:

In[4]:=4
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-s8zp2v
Out[4]=4

Properties & Relations  (12)Properties of the function, and connections to other functions

UpperTriangularMatrixQ returns False for inputs that are not matrices:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-wyx6bq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-l5r7xk
Out[2]=2

Matrices of dimensions {n,0} are upper triangular:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-shucue
Out[1]=1

UpperTriangularize returns matrices that are UpperTriangularMatrixQ:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-59h3h5
Out[1]=1

The inverse of an upper-triangular matrix is upper triangular:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-jg5eek
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-k8yzgl
Out[2]=2

This extends to arbitrary powers and functions:

In[3]:=3
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-jo6a0k
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-tr8i0n
Out[4]=4

The product of two (or more) upper-triangular matrices is upper triangular:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-8103un
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-dwt6ws
Out[2]=2

The determinant of a triangular matrix equals the product of the diagonal entries:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-cmcc5y
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-dt6d6k
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-8nwrmx
Out[3]=3

Eigenvalues of a triangular matrix equal its diagonal elements:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-9bt28q
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-p7rhkx
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-nfcuda
Out[3]=3

UpperTriangularMatrixQ[m,0] is equivalent to UpperTriangularMatrixQ[m]:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-ybjuxk
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-ep5e9m
Out[2]=2

QRDecomposition gives an upper triangular matrix:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-o9w0op
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-bjnbdi
Out[2]=2

CholeskyDecomposition gives an upper triangular matrix:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-khaebf
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-4lqgip
Out[2]=2

HessenbergDecomposition returns a matrix that is upper triangular with an added subdiagonal:

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-y49kqd
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-l4adyx
Out[2]=2

A matrix is upper triangular starting at diagonal iff its transpose is lower triangular starting at diagonal :

In[1]:=1
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-b2gm3j
In[2]:=2
✖
https://wolfram.com/xid/0ixkn43z4qdubwj-2pnb45
Out[2]=2
Wolfram Research (2019), UpperTriangularMatrixQ, Wolfram Language function, https://reference.wolfram.com/language/ref/UpperTriangularMatrixQ.html.
✖
Wolfram Research (2019), UpperTriangularMatrixQ, Wolfram Language function, https://reference.wolfram.com/language/ref/UpperTriangularMatrixQ.html.

Text

Wolfram Research (2019), UpperTriangularMatrixQ, Wolfram Language function, https://reference.wolfram.com/language/ref/UpperTriangularMatrixQ.html.

✖
Wolfram Research (2019), UpperTriangularMatrixQ, Wolfram Language function, https://reference.wolfram.com/language/ref/UpperTriangularMatrixQ.html.

CMS

Wolfram Language. 2019. "UpperTriangularMatrixQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/UpperTriangularMatrixQ.html.

✖
Wolfram Language. 2019. "UpperTriangularMatrixQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/UpperTriangularMatrixQ.html.

APA

Wolfram Language. (2019). UpperTriangularMatrixQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/UpperTriangularMatrixQ.html

✖
Wolfram Language. (2019). UpperTriangularMatrixQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/UpperTriangularMatrixQ.html

BibTeX

@misc{reference.wolfram_2025_uppertriangularmatrixq, author="Wolfram Research", title="{UpperTriangularMatrixQ}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/UpperTriangularMatrixQ.html}", note=[Accessed: 18-May-2025 ]}

✖
@misc{reference.wolfram_2025_uppertriangularmatrixq, author="Wolfram Research", title="{UpperTriangularMatrixQ}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/UpperTriangularMatrixQ.html}", note=[Accessed: 18-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_uppertriangularmatrixq, organization={Wolfram Research}, title={UpperTriangularMatrixQ}, year={2019}, url={https://reference.wolfram.com/language/ref/UpperTriangularMatrixQ.html}, note=[Accessed: 18-May-2025 ]}

✖
@online{reference.wolfram_2025_uppertriangularmatrixq, organization={Wolfram Research}, title={UpperTriangularMatrixQ}, year={2019}, url={https://reference.wolfram.com/language/ref/UpperTriangularMatrixQ.html}, note=[Accessed: 18-May-2025 ]}