Page 292 - Matrix Analysis & Applied Linear Algebra

288 Chapter 5 Norms, Inner Products, and Orthogonality

Norms in Inner-Product Spaces

If V is an inner-product space with an inner product x y , then

= defines a norm on V.

Proof. The fact that = satisfies the first two norm properties in
(5.2.3) on p. 280 follows directly from the defining properties (5.3.1) for an inner
product. You are asked to provide the details in Exercise 5.3.3. To establish the
triangle inequality, use x y ≤| x y | and y x = x y ≤| x y | together
with the CBS inequality to write

2
x + y = x + y x + y = x x + x y + y x + y y
2 2 2
≤ x +2| x y | + y ≤ ( x + y ) .

Example 5.3.2

Problem: Describe the norms that are generated by the inner products pre-
sented in Example 5.3.1.

n×n
• Given a nonsingular matrix A ∈C , the A-norm (or elliptical norm)
n×1
generated by the A-inner product on C is
√
∗
∗
x = x x = x A Ax = Ax . (5.3.5)
A 2
• The standard inner product for matrices generates the Frobenius matrix
norm because

A = A A = trace (A A)= A . (5.3.6)
∗
F
• For the space of real-valued continuous functions defined on (a, b), the norm
b
of a function f generated by the inner product f|g = f(t)g(t)dt is
a
1/2
b
2
f = f|f = f(t) dt .
a