Page 278 - Matrix Analysis & Applied Linear Algebra

274 Chapter 5 Norms, Inner Products, and Orthogonality

There are notions of length other than the euclidean measure. For example,
urban dwellers navigate on a grid of city blocks with one-way streets,so they are
prone to measure distances in the city not as the crow flies but rather in terms
of lengths on a directed grid. For example,instead of than saying that “it’s a
one-half mile straight-line (euclidean) trip from here to there,” they are more
apt to describe the length of the trip by saying,“it’s two blocks north on Dan
Allen Drive,four blocks west on Hillsborough Street,and five blocks south on
Gorman Street.” In other words,the length of the trip is 2 + |− 4| + |− 5| =11
blocks—absolute value is used to insure that southerly and westerly movement
does not cancel the effect of northerly and easterly movement,respectively. This
“grid norm” is better known as the 1-norm because it is a special case of a more
general class of norms defined below.

p-Norms
n n p 1/p
For p ≥ 1, the p-norm of x ∈C is defined as x =( |x i | ) .
p i=1

It can be proven that the following properties of the euclidean norm are in
fact valid for all p-norms:

x ≥ 0 and x =0 ⇐⇒ x = 0,
p p
αx = |α| x for all scalars α, (5.1.7)
p p
x + y ≤ x + y (see Exercise 5.1.13).
p p p
The generalized version of the CBS inequality (5.1.3) for p-norms is H¨older’s
inequality (developed in Exercise 5.1.12),which states that if p> 1 and q> 1
are real numbers such that 1/p +1/q =1, then

∗
|x y|≤ x y . (5.1.8)
p q
In practice,only three of the p-norms are used,and they are
1/2
n n

2
x = |x i | (the grid norm), x = |x i | (the euclidean norm),
1 2
i=1 i=1
and
1/p
n

p
x = lim x p = lim |x i | = max |x i | (the max norm).
∞ i
p→∞ p→∞
i=1
√
For example,if x =(3, 4−3i, 1), then x =9, x = 35, and x =5.
1 2 ∞