Page 489 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Solution

The column vectors of A are

Applying the Gram–Schmidt process with subsequent normalization to these column vectors yields the orthonormal vectors
(see Example 7)

and from 9 the matrix R is

Thus the -decomposition of A is

The Role of the QR-Decomposition in Linear Algebra

In recent years the -decomposition has assumed growing importance as the mathematical foundation for a wide variety of
practical numerical algorithms, including a widely used algorithm for computing eigenvalues of large matrices. Such
algorithms are discussed in textbooks that deal with numerical linear algebra.

Additional Proof

Proof of Theorem 6.3.4 There are two parts to the proof. First we must find vectors and with the stated properties,
and then we must show that these are the only such vectors.

By the Gram–Schmidt process, there is an orthonormal basis for W.

Let

(10)
and