Page 90 - Advanced Linear Algebra
P. 90
74 Advanced Linear Algebra
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1 If =~ : l ; then
:Á; b ; Á: ~
)
2 If ~ :Á; then
im²³ ~ : and ker ²³ ~ ;
and so
=~ im ² ³ l ker ² ³
In other words, is projection onto its image along its kernel. Moreover,
# im ² ³ ¯ # ~#
)
B
3 If ²= ³ has the property that
=~ im ² ³ l ker ² ³ O and ~ im²³
then is projection onto im²³ along ker ²³ .
Projection operators are easy to characterize.
B
Definition A linear operator ²= ³ is idempotent if . ~
Theorem 2.22 A linear operator B ²= ³ is a projection if and only if it is
idempotent.
Proof. If ~ , then for any :Á; : and ! , ;
² b !³~ ~ ~ ² b !³
and so ~ . Conversely, suppose that is idempotent. If # im ² ³ ker ² q , ³
then #~ % and so
~ #~ %~ %~#
Hence im² ³ q ker ² ³ ~ ¸ ¹ . Also, if # = , then
#~²# c #³ b # ker ² ³ l im ² ³
and so =~ ker ² ³ l im ² ³ . Finally, ² %³ ~ % ~ % and so O im ²³ ~ .
Hence, is projection onto im²³ along ker ²³ .
Projections and Invariance
B
Projections can be used to characterize invariant subspaces. Let ²= ³ and
:
:
;
let be a subspace of . Let ~ = :Á; for any complement of . The key is
that the elements of can be characterized as those vectors fixed by , that is,
:
