Page 296 - Matrix Analysis & Applied Linear Algebra
P. 296
292 Chapter 5 Norms, Inner Products, and Orthogonality
Clearly, 2 (p+2)/p =4 only when p =2. Details for the ∞-norm are asked for
in Exercise 5.3.7.
Conclusion: For applications that are best analyzed in the context of an inner-
product space (e.g., least squares problems), we are limited to the euclidean
norm or else to one of its variation such as the elliptical norm in (5.3.5).
n n
Virtually all important statements concerning or C with the standard
inner product remain valid for general inner-product spaces—e.g., consider the
statement and proof of the general CBS inequality. Advanced or more theoretical
texts prefer a development in terms of general inner-product spaces. However,
n n
the focus of this text is matrices and the coordinate spaces and C , so
n n
subsequent discussions will usually be phrased in terms of or C and their
standard inner products. But remember that extensions to more general inner-
product spaces are always lurking in the background, and we will not hesitate
to use these generalities or general inner-product notation when they serve our
purpose.
Exercises for section 5.3
x 1 y 1
5.3.1. For x = x 2 , y = y 2 , determine which of the following are inner
x 3 y 3
3×1
products for .
(a) x y = x 1 y 1 + x 3 y 3 ,
(b) x y = x 1 y 1 − x 2 y 2 + x 3 y 3 ,
(c) x y =2x 1 y 1 + x 2 y 2 +4x 3 y 3 ,
2 2
2 2
2 2
(d) x y = x y + x y + x y .
1 1 2 2 3 3
5.3.2. Fora general inner-product space V, explain why each of the following
statements must be true.
(a) If x y =0 for all x ∈V, then y = 0.
(b) αx y = α x y for all x, y ∈V and for all scalars α.
(c) x + y z = x z + y z for all x, y, z ∈V.
5.3.3. Let V be an inner-product space with an inner product x y . Explain
why the function defined by = satisfies the first two norm
properties in (5.2.3) on p. 280.
2
5.3.4. Fora real inner-product space with = , derive the inequality
2 2
x + y
x y ≤ . Hint: Consider x − y.
2