Page 751 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

9.9 With Gaussian elimination and Gauss–Jordan elimination, a linear system is
solved by operating systematically on an augmented matrix. In this section
-DECOMPOSITIONS we shall discuss a different organization of this approach, one based on
factoring the coefficient matrix into a product of lower and upper triangular
matrices. This method is well suited for computers and is the basis for many
practical computer programs. *

Solving Linear Systems by Factoring

We shall proceed in two stages. First, we shall show how a linear system can be solved very easily once the

coefficient matrix A is factored into a product of lower and upper triangular matrices. Second, we shall show how to

construct such factorizations.

If an matrix A can be factored into a product of matrices as

where L is lower triangular and U is upper triangular, then the linear system can be solved as follows:

Step 1. Rewrite the system as

(1)

Step 2. Define a new matrix y by

(2)

Step 3. Use 2 to rewrite 1 as and solve this system for y.

Step 4. Substitute y in 2 and solve for x.

Although this procedure replaces the problem of solving the single system by the problem of solving the two systems

and , the latter systems are easy to solve because the coefficient matrices are triangular. The following

example illustrates this procedure.

EXAMPLE 1 Solving a System by Factorization
Later in this section we will derive the factorization

Use this result and the method described above to solve the system