Page 338 - Matrix Analysis & Applied Linear Algebra

334 Chapter 5 Norms, Inner Products, and Orthogonality

then
 
x 1
.
 . 
.
 

 x + x  ←− i

2
2
 i j
 
. .
P ij x =  
.
 . 
  ←− j
0
 
 . . 
.
x n
This means that we can selectively annihilate any component—the j th in this
case—by a rotation in the (i, j)-plane without affecting any entry except x i and
x j . Consequently, plane rotations can be applied to annihilate all components
below any particular “pivot.” For example, to annihilate all entries below the
first position in x, apply a sequence of plane rotations as follows:
 √   √   
2
2
2
x +x 2 x +x +x 2 x
1 2 1 2 3
0 0 0
     
 x 3   0   0 
P 12 x=  x 4  , P 13 P 12 x=  x 4  ,. . . , P 1n ···P 13 P 12 x=  0  .
     
. .
.
 .   .   . 
. . .
0
x n x n
The product of plane rotations is generally not another plane rotation, but
such a product is always an orthogonal matrix, and hence it is an isometry. If
n
we are willing to interpret “rotation in ”asa sequence of plane rotations,
then we can say that it is always possible to “rotate” each nonzero vector onto
the first coordinate axis. Recall from (5.6.11) that we can also do this with a
reflection. More generally, the following statement is true.
n
Rotations in
n th
Every nonzero vector x ∈ can be rotated to the i coordinate
axis by a sequence of n − 1 plane rotations. In other words, there is an
orthogonal matrix P such that
Px = x e i , (5.6.17)
where P has the form
P = P in ··· P i,i+1 P i,i−1 ··· P i1 .