 TOPICS # LU Decomposition

A procedure for decomposing an matrix into a product of a lower triangular matrix and an upper triangular matrix , (1)

LU decomposition is implemented in the Wolfram Language as LUDecomposition[m].

Written explicitly for a matrix, the decomposition is (2) (3)

This gives three types of equations (4) (5) (6)

This gives equations for unknowns (the decomposition is not unique), and can be solved using Crout's method. To solve the matrix equation (7)

first solve for . This can be done by forward substitution   (8)   (9)

for , ..., . Then solve for . This can be done by back substitution   (10)   (11)

for , ..., 1.

Gaussian Elimination, Lower Triangular Matrix, Matrix Decomposition, Cholesky Decomposition, QR Decomposition, Triangular Matrix, Upper Triangular Matrix

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## References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "LU Decomposition and Its Applications." §2.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 34-42, 1992.

LU Decomposition

## Cite this as:

Weisstein, Eric W. "LU Decomposition." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LUDecomposition.html