Get the upper triangular part of a matrix:

Get the strictly upper triangular part of a matrix:

Get the upper triangular part of a matrix plus the diagonal below the main diagonal:

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:

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:

Matrices returned by UpperTriangularize satisfy 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:

QRDecomposition gives an upper triangular matrix:

CholeskyDecomposition gives an upper triangular matrix:

JordanDecomposition gives an upper triangular matrix:

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

HermiteDecomposition gives an upper triangular matrix:

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