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:

LUDecomposition decomposes a matrix as a product of upper‐ and lower‐triangular 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:

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:

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

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