Page 290 - Matrix Analysis & Applied Linear Algebra

286 Chapter 5 Norms, Inner Products, and Orthogonality
5.3 INNER-PRODUCT SPACES

The euclidean norm, which naturally came first, is a coordinate-dependent con-
cept. But by isolating its important properties we quickly moved to the more
general coordinate-free definition of a vector norm given in (5.1.9) on p. 275. The
goal is to now do the same for inner products. That is, start with the standard
inner product, which is a coordinate-dependent definition, and identify proper-
ties that characterize the basic essence of the concept. The ones listed below are
those that have been distilled from the standard inner product to formulate a
more general coordinate-free definition.

General Inner Product

An inner product on a real (or complex) vector space V is a function
that maps each ordered pair of vectors x, y to a real (or complex) scalar
x y such that the following four properties hold.
x x is real with x x ≥ 0, and x x =0 if and only if x = 0,
x αy = α x y for all scalars α,
(5.3.1)
x y + z = x y + x z ,
x y = y x (for real spaces, this becomes x y = y x ).
Notice that for each fixed value of x, the second and third properties
say that x y is a linear function of y.

Any real or complex vector space that is equipped with an inner product
is called an inner-product space.

Example 5.3.1

T n×1
∗
• The standard inner products, x y = x y for and x y = x y
n×1
for C , each satisfy the four defining conditions (5.3.1) for a general inner
product—this shouldn’t be a surprise.
∗
∗
• If A n×n is a nonsingular matrix, then x y = x A Ay is an inner product
n×1
for C . This inner product is sometimes called an A-inner product or
an elliptical inner product.
• Consider the vector space of m × n matrices. The functions defined by
T
∗
A B = trace A B and A B = trace (A B) (5.3.2)
m×n m×n
are inner products for and C , respectively. These are referred to
as the standard inner products for matrices. Notice that these reduce
to the standard inner products for vectors when n =1.