Construct an upper triangular matrix:

Show the elements:

Normal can convert an UpperTriangularMatrix to its ordinary representation:

Construct an upper triangular matrix with symbolic entries:

Show the elements:

Get the determinant:

UpperTriangularMatrix objects include properties that give information about the matrix:

The "Summary" property gives a brief summary of information about the matrix:

The "StructuredAlgorithms" property lists the functions that have structured algorithms:

Structured algorithms are typically faster:

Compute the determinant:

Compute the inverse:

Solve a linear system:

Compute the eigenvalues:

When appropriate, structured algorithms return another UpperTriangularMatrix object:

Inverting ut gives another upper triangular matrix:

Transposing ut gives a lower triangular matrix:

The product of ut with its transpose is no longer a triangular matrix:

Construct an upper triangular matrix from a SparseArray with integer entries:

Show the elements:

Construct an upper triangular complex-valued matrix:

Compute the determinant:

Compute the inverse:

Solve a linear system:

Compute the eigenvalues:

LUDecomposition by default uses UpperTriangularMatrix for the upper triangular matrix it returns:

Verify the decomposition:

Format the three matrices:

A function for explicitly computing the Cholesky factor of a Hilbert matrix:

Show that the matrix produced is the same as the result of applying CholeskyDecomposition to the Hilbert matrix:

Verify that multiplying the matrix with its transpose gives the Hilbert matrix:

Elements of the Heisenberg group can be represented by 3×3 upper triangular matrices with unit diagonal:

The product of two Heisenberg matrices is also Heisenberg:

The inverse of a Heisenberg matrix is also Heisenberg:

Transposing an UpperTriangularMatrix yields a LowerTriangularMatrix:

The entries of an UpperTriangularMatrix are coerced to the lowest precision:

A full matrix:

UpperTriangularMatrix[a] does not evaluate if a is not manifestly upper triangular:

Use UpperTriangularMatrix[UpperTriangularize[a]] to get the upper triangular part of a:

An upper triangular matrix with the anti-Hadamard property:

The inverse matrix has integer entries that are unexpectedly large:

The Frobenius norm of the inverse is greater than the Frobenius norm of the original matrix:

Construct a Wilkinson triangular matrix:

It has unit determinant:

The matrix, however, is very ill conditioned: