Let's consider a simple example of LU decomposition using a 2x2 matrix:
Matrix A:
| 2 1 |
| 3 4 |
step:
Further elimination:
Multiply row 2 by -3/2 and add it to row 1. This eliminates the non-zero element in the first column of row 2.
| 2 1 |
| 0 1 |
Previous replacement:
Divide row 2 by 1 (pivot element) to obtain the upper triangular matrix U.
| 2 1 |
| 0 1 |
Build L:
In the original matrix, replace the non-zero elements in the lower triangular part with the ratios used for elimination.
| 1 0 | 0 |
| -3/2 1 | 1 |
Result:
L = | 1 0 |
| -3/2 1 |
u = | 2 1 |
| 0 1 |
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