UpperTriangularMatrixQ

UpperTriangularMatrixQ[m]

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

UpperTriangularMatrixQ[m,k]

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

Details and Options

Examples

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Basic Examples  (3)

Test if a matrix is upper triangular:

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

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

Scope  (12)

Basic Uses  (8)

Test rectangular matrices:

Test a symbolic matrix:

The matrix is upper triangular when c=0:

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

Test a complex matrix:

Test an exact matrix:

Test an arbitrary-precision matrix:

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

Note that this matrix is upper triangular:

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

Note that this matrix is not upper triangular:

Special Matrices  (4)

Test a sparse matrix:

Test structured matrices:

Identity matrices are upper triangular:

Hilbert matrices are not upper triangular:

Options  (1)

Tolerance  (1)

This matrix is not upper triangular:

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

Applications  (3)

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

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

Display the three matrices:

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

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

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

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

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

Visualize the three matrices:

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

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

Properties & Relations  (12)

UpperTriangularMatrixQ returns False for inputs that are not matrices:

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

UpperTriangularize returns matrices that are UpperTriangularMatrixQ:

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

This extends to arbitrary powers and functions:

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

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

Eigenvalues of a triangular matrix equal its diagonal elements:

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

QRDecomposition gives an upper triangular matrix:

CholeskyDecomposition gives an upper triangular matrix:

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

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

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.

CMS

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

BibTeX

@misc{reference.wolfram_2022_uppertriangularmatrixq, author="Wolfram Research", title="{UpperTriangularMatrixQ}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/UpperTriangularMatrixQ.html}", note=[Accessed: 01-July-2022 ]}

BibLaTeX

@online{reference.wolfram_2022_uppertriangularmatrixq, organization={Wolfram Research}, title={UpperTriangularMatrixQ}, year={2019}, url={https://reference.wolfram.com/language/ref/UpperTriangularMatrixQ.html}, note=[Accessed: 01-July-2022 ]}