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5.13 Orthogonal Projection 429
5.13 ORTHOGONAL PROJECTION

As discussed in §5.9, every pair of complementary subspaces defines a projector.
But when the complementary subspaces happen to be orthogonal complements,
the resulting projector has some particularly nice properties, and the purpose of
this section is to develop this special case in more detail. Discussions are in the
context of real spaces, but generalizations to complex spaces are straightforward
by replacing ( ) T by ( ) and “orthogonal matrix” by “unitary matrix.”
∗
⊥
If M is a subspace of an inner-product space V, then V = M⊕M
by (5.11.1), and each v ∈V can be written uniquely as v = m + n, where
⊥
m ∈M and n ∈M by (5.9.3). The vector m was defined on p. 385 to be
the projection of v onto M along M , so the following definitions are natural.
⊥
Orthogonal Projection

For v ∈V, let v = m + n, where m ∈M and n ∈M .
⊥
• m is called the orthogonal projection of v onto M.
⊥
• The projector P M onto M along M is called the orthogonal
projector onto M.
• P M is the unique linear operator such that P M v = m (see p. 386).

3
These ideas are illustrated illustrated in Figure 5.13.1 for V = .

Figure 5.13.1
n
Given an arbitrary pair of complementary subspaces M, N of , formula
(5.9.12) on p. 386 says that the projector P onto M along N is given by

I 0 −1 −1
P = M | N M | N = M | 0 M | N , (5.13.1)
00
where the columns of M and N constitute bases for M and N, respectively.
So, how does this expression simplify when N = M ?To answer the question,
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