If is of rank , then the range of is of dimension . Therefore, it is natural to

de¬ne through its orthogonal complement which has dimension . The above ¥

)

(

¦ ¦

conditions that de¬ne for any become

V

t

©¦ ¤ §¥

¨¦

V

t

¨§V ¨ & ¤ §¥

¦

3

¦

¥¦

These equations de¬ne a projector onto and orthogonal to the subspace . The ¬rst ¥

statement, (1.51), establishes the degrees of freedom, while the second, (1.52), gives

uY u d¥ £ ¡ ¨ ¢¤¥

n ¦¢¥

¤¤§ ¥ §¢

¡

£ ¡© ¥ ¡ © ¥ ¡ ¥ © ¥ ¥

¥ ¨

©

¦

the constraints that de¬ne from these degrees of freedom. The general de¬nition of

projectors is illustrated in Figure 1.1.

¦

¦ ¦

3

¦ ¦

¥¨ ¥

¦

„

U ¥ ¡¢ D

T

¦

Projection of onto and orthogonal to . ¥

The question now is: Given two arbitrary subspaces, and both of dimension , is it ¥

always possible to de¬ne a projector onto orthogonal to through the conditions (1.51) ¥

and (1.52)? The following lemma answers this question.

§ ¥“£

¢§ ©U

T

Given two subspaces and of the same dimension , the following

¥

two conditions are mathematically equivalent.

£

No nonzero vector of is orthogonal to ; ¥

¤£

£

j

‘

¦

For any in there is a unique vector which satis¬es the conditions (1.51)

V

and (1.52).

£

§ § ¥¦ T

The ¬rst condition states that any vector which is in and also orthogonal to ¥

must be the zero vector. It is equivalent to the condition

!7 (

¥¦

¥

˜ 3

Since is of dimension , is of dimension and the above condition is equivalent

¥ ¥ (

to the condition that

t

( ¥ ¤ S ¨ ¢ ¤ §¥

‘ ¦

¦

This in turn is equivalent to the statement that for any , there exists a unique pair of vectors

such that

V

bG’

—

¦

V

Q

V3

¦

where belongs to , and belongs to , a statement which is identical with

¥ (

V

ii.

!7 (

In summary, given two subspaces and , satisfying the condition , there ¥ ¥§

¥

is a projector onto orthogonal to , which de¬nes the projected vector of any vector ¥

V

”

u

¢ £ ¡£ ¡£ ¡

'©

£

¡ £ ¥ ¡ ¤ £ ¦£ § ¡©

¢

¥

©

§

¦

from equations (1.51) and (1.52). This projector is such that

( ¥ C B

C ¢ ¤ )

B )

(

’

)

B

7

¦ ¦ ¦