Page 89 - Advanced Linear Algebra
P. 89
Linear Transformations 73
:
B
=
Definition Let ²= ³ . A subspace of is said to be invariant under or
-invariant
if : : , that is, if : for all : . Put another way, is
:
invariant under if the restriction O is a linear operator on .
: :
If
=~ : l ;
then the fact that is -invariant does not imply that the complement is also
:
;
( s-invariant. The reader may wish to supply a simple example with =~ .)
:
;
B
Definition Let ²= ³ . If = ~ : l ; and if both and are -invariant,
we say that the pair ²:Á ;³ reduces .
A reducing pair can be used to decompose a linear operator into a direct sum as
follows.
B
Definition Let ²= ³ . If ²:Á ;³ reduces we write
~O l O ; :
and call the direct sum of O and O : ; . Thus, the expression
~ l
=
;
:
means that there exist subspaces and of for which : ² Á ; ³ reduces and
~O : and ~O ;
The concept of the direct sum of linear operators will play a key role in the
study of the structure of a linear operator.
Projection Operators
We will have several uses for a special type of linear operator that is related to
direct sums.
Definition Let =~ : l ; . The linear operator :Á; ¢ =¦ = defined by
:Á; ² b !³ ~
where : and ! ; is called projection onto along .
;
:
is a projection, it is with the
Whenever we say that the operator :Á;
understanding that =~ : l ; . The following theorem describes a few basic
properties of projection operators. We leave proof as an exercise.
=
Theorem 2.21 Let be a vector space and let B ² = . ³
