converts the lower triangular matrix lmat to a structured array.
Details and Options
- Lower triangular matrices, when represented as structured arrays, allow for efficient storage and more efficient operations, including Det, Inverse and LinearSolve.
- Lower triangular matrices occur when solving linear systems of equations, where they represent simple systems that can be solved by forward substitution. Matrix decompositions that use lower and upper triangular matrices include LU, LDL, LL (Cholesky) and LQ decompositions.
- A lower triangular matrix satisfies for .
- The elements lij need not be numerical.
- The inverse of a lower triangular matrix is another lower triangular matrix.
- Lower triangular matrices are closed under matrix multiplication, so is again a lower triangular matrix.
- The determinant of a lower triangular matrix is given by the product of the diagonal elements .
- Operations that are accelerated for LowerTriangularMatrix include:
Det time Dot time LinearSolve time
- For a LowerTriangularMatrix sa, the following properties "prop" can be accessed as sa["prop"]:
"Matrix" lower 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[LowerTriangularMatrix[…]] gives the lower triangular matrix as an ordinary matrix.
- LowerTriangularMatrix[…,TargetStructure->struct] returns the lower 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
- LowerTriangularMatrix[…,TargetStructureAutomatic] is equivalent to LowerTriangularMatrix[…,TargetStructure"Structured"].
Examplesopen allclose all
Basic Examples (2)
LowerTriangularMatrix objects include properties that give information about the matrix:
When appropriate, structured algorithms return another LowerTriangularMatrix object:
Construct a lower triangular matrix from a SparseArray with integer entries:
Properties & Relations (2)
Possible Issues (1)
Wolfram Research (2022), LowerTriangularMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/LowerTriangularMatrix.html (updated 2023).
Wolfram Language. 2022. "LowerTriangularMatrix." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/LowerTriangularMatrix.html.
Wolfram Language. (2022). LowerTriangularMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LowerTriangularMatrix.html