¦

¥ ¡¢ U

D T

¦

Orthogonal projection of onto a subspace .

It is interesting to consider the mapping de¬ned as the adjoint of

I

t §¦¥

y x C g B C x I

B ¨

£

©¦

¨ "

¦ ¨ ¦

¨

¦ ¨

First note that is also a projector because for all and ,

I

6 6

C g I % C x B C x % C g I % C x

B

B

I BB B

©¦

¨ ¦ ¨ ¦ ¨ ¦ ¨ ©¦

¨

C

A consequence of the relation (1.60) is

¤ t

)

I B

§)

B( ¨ ¦ §¥¦

(

¤ C t

C

C )

B

( I §0B( ¨ & §¥

¦

C

The above relations lead to the following proposition.

q¤ G S£

£¦

SRFP cbP` Q`

HQ F Q U

T

A projector is orthogonal if and only if it is Hermitian.

C B

£ B¢ ¤

§ § ¥¦ T )

)

(

By de¬nition, an orthogonal projector is one for which . (

C

Therefore, by (1.61), if is Hermitian, then it is orthogonal. Conversely, if is orthogonal,

C

I §¤ §¤

B) )

I B B

B

)

(

0

(

then (1.61) implies while (1.62) implies . Since

C C

C

is a projector and since projectors are uniquely determined by their range and null

P

I

spaces, this implies that . I

Q˜

¨

Given any unitary matrix whose columns form an orthonormal basis of

¤ ¨¨

§()

B

, we can represent by the matrix . This is a particular case of I

C

the matrix representation of projectors (1.57). In addition to being idempotent, the linear

mapping associated with this matrix satis¬es the characterization given above, i.e.,

(

3

B

¨¨ ¨¨

¦ ¦

and

I I

C

It is important to note that this representation of the orthogonal projector is not unique. In

¨

fact, any orthonormal basis will give a different representation of in the above form. As

”

u

¢ £ ¡£ ¡£ ¡

¡

¡ £ ¥ ¡ ¤ £ ¦£ § ¡©

¢

¥

©

§

©

¨0 ¨ ¨ ¨ I0 ¨ 0 ¨

6 I6 6

a consequence for any two orthogonal bases of , we must have ,

an equality which can also be veri¬ed independently; see Exercise 26.

¦ B¨G §¥ ¦ !¢ ¨ C¢¥

¦¢

¡ ¥A

¡ ¢§

¨

2' !GI% §) 3

(

1 §