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P. 265
14.5 QR Decomposition 265
⊥
First, we set v := v 1 . Then
1
1 1 0
1
1
0
v ⊥ := − 2 =
2 2
1 0 1
3 4 1 1 0 1
v 3 ⊥ := − − = −1 .
0
1
1
1 2 0 1 1 0
Then the set
1 0 1
1 , 0 , −1
0 1 0
3
is an orthogonal basis for R . To obtain an orthonormal basis we simply divide each
of these vectors by its length, yielding
1 1
√ 0 √
2
2
1 −1 .
√ , 0 , √
2
2
0 1 0
A 4 × 4 Gram--Schmidt Example
14.5 QR Decomposition
In Chapter 7, Section 7.7 teaches you how to solve linear systems by decom-
posing a matrix M into a product of lower and upper triangular matrices
M = LU .
The Gram–Schmidt procedure suggests another matrix decomposition,
M = QR ,
where Q is an orthogonal matrix and R is an upper triangular matrix. So-
called QR-decompositions are useful for solving linear systems, eigenvalue
problems and least squares approximations. You can easily get the idea
behind the QR decomposition by working through a simple example.
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