Page 92 - Advanced Linear Algebra
P. 92
76 Advanced Linear Algebra
@
?
Example 2.5 Let =~ - and consider the - and -axes and the diagonal:
? ~ ¸²%Á ³ % -¹
@ ~ ¸² Á &³ & -¹
+ ~ ¸²%Á %³ % -¹
Then
+Á? +Á@ ~ +Á@ £ +Á? ~ +Á@ +Á?
From this we deduce that if and are projections, it may happen that both
products and are projections, but that they are not equal. We leave it to
the reader to show that @ Á? ?Á+ ~ (which is a projection), but that ?Á+ @ Á?
is not a projection.
Since a projection is idempotent, we can write the identity operator as s sum
of two orthogonal projections:
b ² c³ ~ Á ² c³
Let us generalize this to more than two projections.
Definition A resolution of the identity on is a sum of the form
=
bÄb ~
where the 's are pairwise orthogonal projections, that is, for £ .
There is a connection between the resolutions of the identity on and direct
=
sum decompositions of . In general terms, if
=
bÄb ~
B
for any linear operators ²= ³ , then for all # = ,
#~ # b Ä b # im ² ³ b Ä b im ² ³
and so
=~ im ² ³ b Ä b im ² ³
However, the sum need not be direct.
Theorem 2.25 Let = be a vector space. Resolutions of the identity on =
correspond to direct sum decompositions of as follows:
=
) bÄb ~ is a resolution of the identity, then
1 If
=~ im ² ³ l Ä l im ² ³
