LowerTriangularize

LowerTriangularize[m]

gives a matrix in which all but the lower triangular elements of m are replaced with zeros.

LowerTriangularize[m,k]

replaces with zeros only the elements above the k^(th) subdiagonal of m.

Details and Options

Examples

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Basic Examples  (3)

Get the lower triangular part of a matrix:

Get the strictly lower triangular part of a matrix:

Get the lower triangular part of a matrix plus the diagonal above the main diagonal:

Scope  (12)

Basic Uses  (8)

Get the lower triangular part of nonsquare matrices:

Find the lower triangular part of a machine-precision matrix:

Lower triangular part of a complex matrix:

Lower triangular part of an exact matrix:

Lower triangular part of an arbitrary-precision matrix:

Compute the lower triangular part of a symbolic matrix:

Large matrices are handled efficiently:

The number of rows or columns limits the meaningful values of the parameter k:

Special Matrices  (4)

The lower triangular part of a sparse matrix is returned as a sparse matrix:

Format the result:

The lower triangular part of structured matrices:

The lower triangular part of an identity matrix is the matrix itself:

This is true of any diagonal matrix:

Compute the lower triangular part, including the superdiagonal, for HilbertMatrix:

Options  (2)

TargetStructure  (2)

A matrix:

Return the result as a dense matrix:

Return the result as a sparse matrix:

Return the result as a LowerTriangularMatrix:

A sparse array:

The setting TargetStructureAutomatic gives a sparse result:

Convert the sparse array to a dense matrix:

The setting TargetStructureAutomatic gives a dense result:

Applications  (2)

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

Extract the strictly lower part of lu with LowerTriangularize and place ones on the diagonal:

Extract the upper part of lu with UpperTriangularize:

Display the three matrices:

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

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  (6)

Matrices returned by LowerTriangularize satisfy LowerTriangularMatrixQ:

The inverse of a lower triangular matrix is lower triangular:

This extends to arbitrary powers and functions:

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

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

Eigenvalues of a triangular matrix equal its diagonal elements:

LowerTriangularize[m,k] is equivalent to Transpose[UpperTriangularize[Transpose[m], -k]]:

Wolfram Research (2008), LowerTriangularize, Wolfram Language function, https://reference.wolfram.com/language/ref/LowerTriangularize.html (updated 2023).

Text

Wolfram Research (2008), LowerTriangularize, Wolfram Language function, https://reference.wolfram.com/language/ref/LowerTriangularize.html (updated 2023).

CMS

Wolfram Language. 2008. "LowerTriangularize." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/LowerTriangularize.html.

APA

Wolfram Language. (2008). LowerTriangularize. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LowerTriangularize.html

BibTeX

@misc{reference.wolfram_2024_lowertriangularize, author="Wolfram Research", title="{LowerTriangularize}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/LowerTriangularize.html}", note=[Accessed: 24-May-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_lowertriangularize, organization={Wolfram Research}, title={LowerTriangularize}, year={2023}, url={https://reference.wolfram.com/language/ref/LowerTriangularize.html}, note=[Accessed: 24-May-2024 ]}