Page 248 - Advanced Linear Algebra
P. 248
232 Advanced Linear Algebra
Proof. Since
² ³ i ~ :Á; ;Á:
it follows that ~ i if and only if : ~ ; , that is, if and only if is
orthogonal. Hence, 1) and 2) are equivalent.
To prove that 1) implies 3), let ~ . Then if # ~ :Á: b ! for : and
!: , it follows that
) ) ~ )) b ) ) ! )) ~ ) # )
#
Now suppose that 3) holds. Then
im²³ l ker ²³ ~ = ~ ker ²³ p ker ²³
and we wish to show that the first sum is orthogonal. If $ im ² ³ , then
$ ~ % b & , where % ² ker ³ & and ² ker ³ . Hence,
$~ $~ % b & ~ &
and so the orthogonality of and implies that
%
&
&
$
) ) b )) ~ & ) ) ~ ) & ) ))
%
Hence, % ~ and so im ²³ ker ²³ , which implies that im ²³ ~ ker ²³ .
Orthogonal Resolutions of the Identity
We have seen (Theorem 2.25) that resolutions of the identity
bÄb ~
on = correspond to direct sum decompositions of = . If, in addition, the
projections are orthogonal, then the direct sum is an orthogonal sum.
Definition An orthogonal resolution of the identity is a resolution of the
identity bÄb ~ in which each projection is orthogonal.
The following theorem displays a correspondence between orthogonal direct
sum decompositions of and orthogonal resolutions of the identity.
=
Theorem 10.6 Let be an inner product space. Orthogonal resolutions of the
=
identity on = correspond to orthogonal direct sum decompositions of = as
follows:
) bÄb ~ is an orthogonal resolution of the identity, then
1 If
=~ im ² ³ p Ä p im ² ³
and is orthogonal projection onto im²³ .
