converts the upper triangular matrix umat to a structured array.
Details and Options
- Upper triangular matrices, when represented as structured arrays, allow for efficient storage and more efficient operations, including Det, Inverse and LinearSolve.
- Upper triangular matrices occur when solving linear systems of equations, where they represent simple systems that can be solved by back substitution. Matrix decompositions that use upper and lower triangular matrices include LU, UU (Cholesky) and QR decompositions.
- An upper triangular matrix satisfies for .
- The elements uij need not be numerical.
- The inverse of an upper triangular matrix is also upper triangular.
- Upper triangular matrices are closed under matrix multiplication, so is again an upper triangular matrix.
- The determinant of an upper triangular matrix is given by the product of the diagonal elements .
- Operations that are accelerated for UpperTriangularMatrix include:
Det time Dot time LinearSolve time
- For an UpperTriangularMatrix sa, the following properties "prop" can be accessed as sa["prop"]:
"Matrix" upper triangular matrix, represented as a full array "Properties" list of supported properties "Structure" type of structured array "StructuredData" internal data stored by the structured array "StructuredAlgorithms" list of functions with special methods for the structured array "Summary" summary information, represented as a Dataset
- Normal[UpperTriangularMatrix[…]] gives the upper triangular matrix as an ordinary matrix.
- UpperTriangularMatrix[…,TargetStructure->struct] returns the upper triangular matrix in the format specified by struct. Possible settings include:
Automatic automatically choose the representation returned "Dense" represent the matrix as a dense matrix "Sparse" represent the matrix as a sparse array "Structured" represent the matrix as a structured array
- UpperTriangularMatrix[…,TargetStructureAutomatic] is equivalent to UpperTriangularMatrix[…,TargetStructure"Structured"].
Examplesopen allclose all
Basic Examples (2)
UpperTriangularMatrix objects include properties that give information about the matrix:
When appropriate, structured algorithms return another UpperTriangularMatrix object:
Construct an upper triangular matrix from a SparseArray with integer entries:
Show that the matrix produced is the same as the result of applying CholeskyDecomposition to the Hilbert matrix:
Properties & Relations (2)
Possible Issues (1)
Neat Examples (2)
Wolfram Research (2022), UpperTriangularMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/UpperTriangularMatrix.html (updated 2023).
Wolfram Language. 2022. "UpperTriangularMatrix." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/UpperTriangularMatrix.html.
Wolfram Language. (2022). UpperTriangularMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/UpperTriangularMatrix.html