A procedure for decomposing an matrix into a product of a lower triangular matrix and an upper triangular matrix ,

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LU decomposition is implemented in Mathematica as LUDecomposition[m].

Written explicitly for a matrix, the decomposition is

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This gives three types of equations

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j l_(i1)u_(1j)+l_(i2)u_(2j)+...+l_(ij)u_(jj)=a_(ij). " width="240" border="0" height="18">	(6)

This gives equations for unknowns (the decomposition is not unique), and can be solved using Crout's method. To solve the matrix equation

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first solve for . This can be done by forward substitution

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for , ..., . Then solve for . This can be done by back substitution

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for , ..., 1.

Related Keyword:Gaussian Elimination, Lower Triangular Matrix, Matrix Decomposition, Cholesky Decomposition, QR Decomposition, Triangular Matrix, Upper Triangular Matrix