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:

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:

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 :

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:

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 :