¥

(

C

The converse is also true. Hence, the following useful property,

t

‚

¥ ¨ £ ¤ ¦§¥

a

7

¦ ¦

iff ¦

¦ ¦

¤¥A ¦¢

A¡ ¡ ' 6

§ !PG ' ¨¢ ¨ ¥ ¥§

§¨

Two bases are required to obtain a matrix representation of a general projector: a basis

¤

”¥ )1)1 0 1 )11) 0 1

3 3

B¢)

¨ (

for the subspace and a second one ¡ ”

C

for the subspace . These two bases are biorthogonal when

¨ ¤ ¤ t ¦§¥

‘H…1 C v C B

… a

¨

¨

¦ ¨

In matrix form this means . Since belongs to , let be its representation

¡ I

3

¨ ¦ ¦

in the basis. The constraint is equivalent to the condition,

¥¦

C v C ¨

…

i)1)‘}

BB 3 7

"

¦ ¨

for

In matrix form, this can be rewritten as

t

z C ¨ BI ¡

¨ ¤ ¦§¥

3 8

7

¦ ¨

I¡ ¨ ¦

If the two bases are biorthogonal, then it follows that . Therefore, in this case,

‚

¢¨

¦ ¦

, which yields the matrix representation of ,

¡ I

t

I ¢¨ ¨ ¤ ¦§¥

(

¡

¨

In case the bases and are not biorthogonal, then it is easily seen from the condition

¡

(1.56) that

t

¨

0 ¨ ¤ ¦§¥

¡B 2

¨ ¡

I I

&

C

‚

If we assume that no vector of is orthogonal to , then it can be shown that the ¥

¨

matrix is nonsingular.¡ I

¥¤¥A ¦¢ ¦ !¢ ¨ ¢¥

# ¦ £¤H©G' 2'0)GI !

¨ ¥1

¡¡ (% §

1 C§

§

An important class of projectors is obtained in the case when the subspace is equal to

¥

, i.e., when

(C § ¤ C B

) B 0

(

Then, the projector is said to be the orthogonal projector onto . A projector that is

not orthogonal is oblique. Thus, an orthogonal projector is de¬ned through the following

¦

requirements satis¬ed for any vector ,

¤ t

() ¨ ¢ ¤ §¥

3B ¦

(

¦ ¦

¦

C

uY u d¥ £ ¡ ¨ ¢¤¥

n ¦¢¥

¤¤§ ¥ §¢

¡

£ ¡© ¥ ¡ © ¥ ¡ ¥ © ¥ ¥

¥ ¨

©

or equivalently,

¤ C xC

( ()

3 BB

78

¦ ¨© ¦ ¨

¦

¦

¦

3