Page 274 - Matrix Analysis & Applied Linear Algebra

270 Chapter 5 Norms, Inner Products, and Orthogonality

2 3
is called the euclidean norm in and , and there is an obvious extension
to higher dimensions.

Euclidean Vector Norm
Foravector x n×1 , the euclidean norm of x is defined to be

1/2 √
n 2 n×1
T
• x = x = x x whenever x ∈ ,
i=1 i
1/2 √
n 2 n×1
∗
• x = |x i | = x x whenever x ∈C .
i=1
   
0 i
−1 2
   
For example, if u =  2  and v =  1 − i , then
−2 0
4 1+i
√
√
2
T
u = u = u u = 0+1+4+4+16 = 5,
i

√ √
2
v = |v i | = v v = 1+4+2+0+2 = 3.
∗
33
There are several points to note.
• The complex version of x includes the real version as a special case because
2
|z| = z 2 whenever z is a real number. Recall that if z = a +ib, then
√ √
2
2
¯ z = a − ib, and the magnitude of z is |z| = ¯ zz = a + b . The fact that
2
2
2
|z| =¯zz = a + b is a real number insures that x is real even if x has
some complex components.
• The definition of euclidean norm guarantees that for all scalars α,
x ≥ 0, x =0 ⇐⇒ x = 0, and αx = |α| x . (5.1.1)
• Given a vector x = 0, it’s frequently convenient to have another vector
that points in the same direction as x (i.e., is a positive multiple of x) but
has unit length. To construct such a vector, we normalize x by setting
u = x/ x . From (5.1.1), it’s easy to see that

x
1
u =

= x =1. (5.1.2)

x x
33
By convention, column vectors are used throughout this chapter. But there is nothing special
about columns because, with the appropriate interpretation, all statements concerning columns
will also hold for rows.