generates a representation of the LU decomposition of a square matrix m.
Details and Options
- LUDecomposition returns a list of three elements. The first element is a combination of upper‐ and lower‐triangular matrices, the second element is a vector specifying rows used for pivoting, and for approximate numerical matrices m the third element is an estimate of the L∞ condition number of m.
Examplesopen allclose all
Basic Examples (2)
Basic Uses (7)
Use LUDecomposition with an exact matrix:
Use LUDecomposition with a symbolic matrix:
Special Matrices (4)
Illustrate the structure by using MatrixPlot:
A triangular linear system is a system of linear equations in which the first equation has one variable and each subsequent equation introduces exactly one additional variable. Rewrite the following system in four variables as two triangular linear systems in eight variables:
LinearSolve[m] sets up an LU decomposition in a functional form convenient for solving:
This can be done manually with the output of LUDecomposition as well:
The sign can be fixed using Signature[p]:
Properties & Relations (9)
The permutation list returned by LUDecomposition can be converted to a matrix as follows:
The CholeskyDecomposition of a positive-definite, Hermitian matrix h:
This gives a kind of LU decomposition via ConjugateTranspose:
This is generally a different decomposition from the one given by LUDecomposition:
The value returned by LUDecomposition is only an estimate:
The condition number is the same condition number reported by LinearSolve[m]
Wolfram Research (1996), LUDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/LUDecomposition.html.
Wolfram Language. 1996. "LUDecomposition." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LUDecomposition.html.
Wolfram Language. (1996). LUDecomposition. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LUDecomposition.html