Page 91 - Advanced Linear Algebra
P. 91
Linear Transformations 75
: if and only if ~ . Hence, the following are equivalent:
: :
: for all :
² ³ ~ for all :
³ ~ for all :
²
Thus, is -invariant if and only if ~ for all vectors : . But this is
:
also true for all vectors in , since both sides are equal to on . This proves
;
;
the following theorem.
=
:
Theorem 2.23 Let ²= ³ . Then a subspace of is -invariant if and only
B
if there is a projection ~ :Á; for which
~
in which case this holds for all projections of the form ~ :Á; .
We also have the following relationship between projections and reducing pairs.
Theorem 2.24 Let =~ : l ; . Then ²:Á ;³ reduces ²= ³ if and only if
B
.
commutes with :Á;
Proof. Theorem 2.23 implies that and are -invariant if and only if
;
:
:Á; :Á; ~ :Á; and ²c :Á; ³ ²c :Á; ³ ~ ²c :Á; ³
and a little algebra shows that this is equivalent to
:Á; :Á; ~ :Á; and :Á; ~ :Á;
which is equivalent to :Á; ~ :Á; .
Orthogonal Projections and Resolutions of the Identity
Observe that if is a projection, then
³
²c ³ ~ ²c ~
Definition Two projections Á B = ² ³ are orthogonal , written , if
~ ~
Note that if and only if
im²³ ker ² ³ im² ³ ker ²³
and
The following example shows that it is not enough to have ~ in the
definition of orthogonality. In fact, it is possible for ~ and yet is not
even a projection.
