Page 348 - Matrix Analysis & Applied Linear Algebra

344 Chapter 5 Norms, Inner Products, and Orthogonality

Elementary reflectors are not the only type of orthogonal matrices that can
be used to reduce a matrix to an upper-trapezoidal form. Plane rotation matrices
are also orthogonal, and, as explained on p. 334, plane rotation matrices can be
used to selectively annihilate any component in a given column, so a sequence of
plane rotations can be used to annihilate all elements below a particular pivot.
m×n
This means that a matrix A ∈ can be reduced to an upper-trapezoidal
form strictly by using plane rotations—such a process is usually called a Givens
reduction.
Example 5.7.2

Problem: Use Givens reduction (i.e., use plane rotations) to reduce the matrix
 
0 −20 −14
A =  3 27 −4 
4 11 −2
to upper-triangular form. Also compute an orthogonal matrix P such that
PA = T is upper triangular.
Solution: The plane rotation that uses the (1,1)-entry to annihilate the (2,1)-
entry is determined from (5.6.16) to be

   
010 327 −4
P 12 =  −100  so that P 12 A =  020 14  .
001 411 −2
Now use the (1,1)-entry in P 12 A to annihilate the (3,1)-entry in P 12 A. The
plane rotation that does the job is again obtained from (5.6.16) to be
   
304 5 25 −4
1
P 13 =  050  so that P 13 P 12 A =  0 20 14  .
5
−403 0 −15 2
Finally, using the (2,2)-entry in P 13 P 12 A to annihilate the (3,2)-entry produces
   
50 0 525 −4
1
P 23 =  04 −3  so that P 23 P 13 P 12 A = T =  025 10  .
5
03 4 0 0 10
Since plane rotation matrices are orthogonal, and since the product of orthogonal
matrices is again orthogonal, it must be the case that
 
0 15 20
1
P = P 23 P 13 P 12 =  −20 12 −9 
25
−15 −16 12
is an orthogonal matrix such that PA = T.