Page 253 - A Course in Linear Algebra with Applications

7.3: Orthonormal Sets and the Gram-Schmidt Process 237

We remark that these matrices have already appeared
in other contexts. The first matrix represents an anticlock-
wise rotation in R 2 through angle 0: see Example 6.2.6. The
second matrix corresponds to a reflection in R 2 in the line
through the origin making angle 0/2 with the positive IE-
direction; see Exercises 6.2.3 and 6.2.6. Thus a connection
2
has been established between x2 real orthogonal matrices on
the one hand, and rotations and reflections in 2-dimensional
Euclidean space on the other.
It is worthwhile restating the QR-factorization principle
in the important case where the matrix A is invertible.
Theorem 7.3.6
Every invertible real matrix A can be written as a product QR
where Q is a real orthogonal matrix and R is a real upper tri-
angular matrix with positive entries on its principal diagonal.
Example 7.3.8
Write the following matrix in the QR-factorized form:

A

The method is to apply the Gram-Schmidt process to
the columns X\, X 2, X3 of A, which are linearly independent
and so form a basis for the column space of A. This yields an
orthonormal basis {Yi, Y 2, Y 3} where

^ = -1 n=^,

y 2 +
-;*UJ- T* T*