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: