Page 440 - Matrix Analysis & Applied Linear Algebra

436 Chapter 5 Norms, Inner Products, and Orthogonality

In other words, min m∈M b − m = b − p . Now argue that there is not
2 2
another point in M that is as close to b as p is. If : m ∈M such that
b − : m = b − p , then by using the Pythagorean theorem again we see
2 2
2 2 2 2
b − : m = b − p + p − : m = b − p + p − : m =⇒ p − : m =0,
2 2 2 2 2
and thus : m = p.
Example 5.13.4
n×n
To illustrate some of the previous ideas, consider with the inner product
T
A B = trace A B . If S n is the subspace of n × n real-symmetric matrices,
then each of the following statements is true.
⊥
• S n = the subspace K n of n × n skew-symmetric matrices.
S n ⊥K n because for all S ∈S n and K ∈K n ,
T T T T
S K = trace S K = −trace SK = −trace SK
T T
= −trace KS = −trace S K = − S K
=⇒ S K =0.

n×n n×n
= S n ⊕K n because every A ∈ can be uniquely expressed
as the sum of a symmetric and a skew-symmetric matrix by writing
A + A T A − A T
A = + (recall (5.9.3) and Exercise 3.2.6).
2 2
n×n T
• The orthogonal projection of A ∈ onto S n is P(A)=(A + A )/2.
n×n T
• The closest symmetric matrix to A ∈ is P(A)=(A + A )/2.
n×n
• The distance from A ∈ to S n (the deviation from symmetry) is

2
T

T
trace (A A)−trace (A )
dist(A, S n )= A−P(A) = (A−A )/2
= .

F F 2
Example 5.13.5
Affine Projections. If v = 0 is a vector in a space V, and if M is a
subspace of V, then the set of points A = v + M is called an affine space in
V. Strictly speaking, A is not a subspace (e.g., it doesn’t contain 0 ), but, as
depicted in Figure 5.13.5, A is the translate of a subspace—i.e., A is just a copy
of M that has been translated away from the origin through v. Consequently,
notions such as projection onto A and points closest to A are analogous to the
corresponding concepts for subspaces.