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Home > Matrix & Vector calculators > LU Decomposition using Gauss Elimination method of Matrix example
8. LU decomposition using Gauss Elimination method of matrix example ( Enter your problem )
( Enter your problem )
  1. Example `[[8,-6,2],[-6,7,-4],[2,-4,3]]`
  2. Example `[[3,2,4],[2,0,2],[4,2,3]]`
  3. Example `[[1,1,1],[-1,-3,-3],[2,4,4]]`
  4. Example `[[2,3],[4,10]]`
 		Other related methods
  1. Transforming matrix to Row Echelon Form
  2. Transforming matrix to Reduced Row Echelon Form
  3. Rank of matrix
  4. Characteristic polynomial of matrix
  5. Eigenvalues
  6. Eigenvectors (Eigenspace)
  7. Triangular Matrix
  8. LU decomposition using Gauss Elimination method of matrix
  9. LU decomposition using Doolittle's method of matrix
  10. LU decomposition using Crout's method of matrix
  11. Diagonal Matrix
  12. Cholesky Decomposition
  13. QR Decomposition (Gram Schmidt Method)
  14. QR Decomposition (Householder Method)
  15. LQ Decomposition
  16. Pivots
  17. Singular Value Decomposition (SVD)
  18. Moore-Penrose Pseudoinverse
  19. Power Method for dominant eigenvalue
  20. determinants using Sarrus Rule
  21. determinants using properties of determinants
  22. Row Space
  23. Column Space
  24. Null Space
2. Example `[[3,2,4],[2,0,2],[4,2,3]]`
(Previous example)		4. Example `[[2,3],[4,10]]`
(Next example)

3. Example `[[1,1,1],[-1,-3,-3],[2,4,4]]`


Find LU Decomposition using Gauss Elimination method of Matrix ...
`[[1,1,1],[-1,-3,-3],[2,4,4]]`

Solution:
`LU` decomposition : If we have a square matrix A, then an upper triangular matrix U can be obtained without pivoting under Gaussian Elimination method, and there exists lower triangular matrix L such that A=LU.


Here `A` = 
`1``1``1`
`-1``-3``-3`
`2``4``4`


Using Gaussian Elimination method
`R_2 larr R_2-``(-1)``xx R_1` `[:.L_(2,1)=color{blue}{-1}]`

 = 
`1``1``1`
`0``-2``-2`
`2``4``4`


`R_3 larr R_3-``(2)``xx R_1` `[:.L_(3,1)=color{blue}{2}]`

 = 
`1``1``1`
`0``-2``-2`
`0``2``2`


`R_3 larr R_3-``(-1)``xx R_2` `[:.L_(3,2)=color{blue}{-1}]`

 = 
`1``1``1`
`0``-2``-2`
`0``0``0`


`:.U` = 
`1``1``1`
`0``-2``-2`
`0``0``0`


`L` is just made up of the multipliers we used in Gaussian elimination with 1s on the diagonal.

`:.L` = 
`1``0``0`
`color{blue}{-1}``1``0`
`color{blue}{2}``color{blue}{-1}``1`


`:.` LU decomposition for A is

`A` = 
`1``1``1`
`-1``-3``-3`
`2``4``4`
 = 
`1``0``0`
`-1``1``0`
`2``-1``1`
 `xx` 
`1``1``1`
`0``-2``-2`
`0``0``0`
 = `LU`


This material is intended as a summary. Use your textbook for detail explanation.
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2. Example `[[3,2,4],[2,0,2],[4,2,3]]`
(Previous example)		4. Example `[[2,3],[4,10]]`
(Next example)


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