Page 315 - Matrix Analysis & Applied Linear Algebra

5.5 Gram–Schmidt Procedure 311

The factorization A = QR is called the QR factorization for A, and it is
uniquely determined by A (Exercise 5.5.8). When A and Q are not square,
some authors emphasize the point by calling A = QR the rectangular QR
factorization—the case when A and Q are square is further discussed on p. 345.
Below is a summary of the above observations.

QR Factorization
Every matrix A m×n with linearly independent columns can be uniquely
factored as A = QR in which the columns of Q m×n are an orthonor-
mal basis for R (A) and R n×n is an upper-triangular matrix with
positive diagonal entries.

• The QR factorization is the complete “road map” of the Gram–

are
Schmidt process because the columns of Q = q 1 | q 2 |···| q n
the result of applying the Gram–Schmidt procedure to the columns

and R is given by
of A = a 1 | a 2 |···| a n
 ∗ ∗ ∗ 
ν 1 q a 2 q a 3 ··· q a n
1
1
1
∗
∗
 0 ν 2 q a 3 ··· q a n 


2
2
 
∗
 0 0 ν 3 ··· q a n 
3
R =   ,
 . . . . . . . . . . 


 . . . . . 
0 0 0 ··· ν n

k−1

where ν 1 = a 1 and ν k = a k − q i a k q i
for k> 1.
i=1
Example 5.5.2
Problem: Determine the QR factors of
 
0 −20 −14
A =  3 27 −4  .
4 11 −2
n
Solution: Using the standard inner product for , apply the Gram–Schmidt
procedure to the columns of A by setting
k−1 T
a 1 a k − i=1 q a k q i
i
q 1 = and q k = for k =2, 3,
ν 1 ν k

k−1 T

where ν 1 = a 1 and ν k = a k − q a k q i . The computation of these
i=1 i
quantities can be organized as follows.