Page 265 - A Course in Linear Algebra with Applications

7.4: The Method of Least Squares 249

A = QR, as in 7.3.5. Thus Q is an m x n matrix with or-
thonormal columns and R is an n x n upper triangular matrix
with positive diagonal elements. Since the columns of Q form
T
an orthonormal set, Q Q = I n. Hence
T
T
T
T
A A = R Q QR = R R.
T
1 T
T
Thus X = {R R)- R Q B, which reduces to
X T
X = R- Q B,
a considerable simplification of the original formula. However
Q and R must already be known before this formula can be
used.

Example 7.4.3
Consider the least squares problem

Xi + x 2 + 2x 3 = 1

Xi + 2x 2 + 3x 3 = 2
+
Xi + 2x 2 x 3 = 1
Here
' \ 1 2\ (I
A = 2 3 and B = 1
2 l) V

It was shown in Example 7.3.8 that A = QR where

l/>/3 -1/V6 1/
1/V2'
Q=' ' 1/V3 2/v^ 0
1/V3 - l / \ / 6 - l / \ / 2 /

and
V^ 4/\/3 2 ^ \
R 0 >/6/3 \/6/2
0 0 V ^ / 2 /