3B
¦ ¦
When is an orthogonal projector, then the two vectors and in the decom
‚
C
— B 3
¦ ¦ ¦
position are orthogonal. The following relation results:
6 ¦ C 6 6
— 3B
6 6 6
¦ ¦
C
¦
A consequence of this is that for any ,
A6
#6
¦ ¦
6 ¦ (6
j
‘
¦ ¦
Thus, the maximum of , for all in does not exceed one. In addition the
¤
¢)
B(
value one is reached for any element in . Therefore,
C –6
for any orthogonal projector .
An orthogonal projector has only two eigenvalues: zero or one. Any vector of the range
of is an eigenvector associated with the eigenvalue one. Any vector of the nullspace is
obviously an eigenvector associated with the eigenvalue zero.
Next, an important optimality property of orthogonal projectors is established.
y SG b¡¤
§¥¦ ¥ Q T
Let be the orthogonal projector onto a subspace . Then for any
j
‘
¦
given vector in , the following is true:
©¢ t
X6
A6
( ¨ ¢ §¥
¦
3 3
¦ ¨ ¦ ¦
¡¢
£
©¨¦
§ §¥ T
¨ ¦
Let be any vector of and consider the square of its distance from . Since
3 3
¦ ¦ ¦ ¨
is orthogonal to to which belongs, then
66 66 C 6 66 C
— —
B B
3 3 3 3 3
6
¦ ¨ ¦ ¦ ¦ ¨ ¦ ¦ ¦ ¨
( 6
6
3 ¦¦
3
¦ ¨ ¦ ¨
Therefore, for all in . This establishes the result by noticing
¨ ¦
that the minimum is reached for .
2
¨ ¦
By expressing the conditions that de¬ne for an orthogonal projector onto
Y
a subspace , it is possible to reformulate the above result in the form of necessary and
suf¬cient conditions which enable us to determine the best approximation to a given vector
¦
in the leastsquares sense.
¡b£ £ G
¦
¦¢
`Q
Q ©U
T
–
‘
¦
Let a subspace , and a vector in be given. Then
©¢ 62 t
6
( ¨ £ §¥
¦
3 3
¦ ¨ ¦ ¨
¡¢
if and only if the following two conditions are satis¬ed,
2
£
¨
2
3
¦ ¨
¦
uY u d¥ £ ¡ ¨ ¢¤¥
n ¦¢¥
© ¤¤§ ¥ §¢
¡
£ ¡© ¥ ¡ © ¥ ¡ ¥ © ¥ ¥
¥ ¨
$ ¨ "
¨ ¨ £ ©§ £ $( ) " ( ) § § £
) )
)
%
& 1{˜{

Linear systems are among the most important and common problems encountered in scien
ti¬c computing. From the theoretical point of view, the problem is rather easy and explicit
solutions using determinants exist. In addition, it is well understood when a solution ex