LU decomposition for a 3x3 matrix
To perform LU decomposition of a 3x3 matrix step by step, we decompose matrix into a lower triangular matrix and an upper triangular matrix such that:
Where:
- is a lower triangular matrix, meaning it has non-zero values only on and below the diagonal, and all elements above the diagonal are zero.
- is an upper triangular matrix, meaning it has non-zero values only on and above the diagonal, and all elements below the diagonal are zero.
Given:
We need to find:
General procedure:
For each , we follow these two steps:
(a) Use the following equation to solve for each where (in the below equation, the summation term is zero when ):
(b) Use the following equation to solve for where :
(a) Solve for where
(b) Solve for where (, as already known):
(a) Solve for where
(b) Solve for where (, as already known):
(a) Solve for where
(b) Solve for where (, as already known)
Numerical example
Let's consider:
(a) Solve for where
(b) Solve for where (, as already known):
(a) Solve for where
(b) Solve for where (, as already known):
(a) Solve for where
Thus, the matrices and are: