Suppose we have the system of equations The motivation for an

Suppose we have the system of equations

The motivation for an decomposition is based on the observation that systems of equations involving triangular coefficient matrices are easier to deal with. Indeed, the whole point of Gaussian elimination is to replace the coefficient matrix with one that is triangular. The decomposition is another approach designed to exploit triangular systems.
We suppose that we can write

where is a lower triangular matrix and is an upper triangular matrix. Our aim is to find and and once we have done so we have found an decomposition of .
Key Point 5
An decomposition of a matrix is the product of a lower triangular matrix and an upper triangular matrix that is equal to .
It turns out that we need only consider lower triangular matrices that have 1 s down the diagonal. Here is an example. Let

Multiplying out and setting the answer equal to gives