# LowerTriangularMatrixQ

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

gives True if m is lower triangular starting down from the k diagonal, and False otherwise.

# Details and Options • works even if m is not a square matrix.
• In , positive k refers to superdiagonals above the main diagonal and negative k refers to subdiagonals below the main diagonal.
• LowerTriangularMatrixQ works with SparseArray and structured array objects.
• The following option can be given:
•  Tolerance Automatic tolerance for approximate numbers
• For approximate matrices, the option Tolerance->t can be used to indicate that all entries Abs[mij]t are taken to be zero.

# Examples

open allclose all

## Basic Examples(3)

Test if a matrix is lower triangular:

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

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

## Scope(12)

### Basic Uses(8)

Test rectangular matrices:

Test a symbolic matrix:

The matrix is lower triangular when b=0:

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

Test if a complex matrix is lower triangular:

Test if an exact matrix is lower triangular:

Test an arbitrary-precision matrix:

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

Note that this matrix is not lower triangular:

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

Note that this matrix is lower triangular:

### Special Matrices(4)

Test a sparse matrix:

Test structured matrices:

Identity matrices are lower triangular:

Hilbert matrices are not lower triangular:

## Options(1)

### Tolerance(1)

This matrix is not lower triangular:

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

## Applications(2)

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:

JordanDecomposition relates any matrix to an upper-triangular matrix via a similarity transformation :

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(9)

LowerTriangularMatrixQ returns False for inputs that are not matrices:

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

LowerTriangularize returns matrices that are 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:

is equivalent to :

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