Page 244 - Advanced Linear Algebra
P. 244
228 Advanced Linear Algebra
Specifically, for each $> , the linear functional = i defined by
$
# ~ º #Á $»
$
has the form
# ~ º#Á 9 »
$
$
where 9 = is the Riesz vector for $ $ . If ¢ i > ¦ = is defined by
i
$~9 $ $ ~9² ³
where is the Riesz map, then
9
º#Á i $» ~ º#Á 9 » ~ # ~ º #Á $»
$ $
i
9
Finally, since ~9 k is the composition of the Riesz map and the map
¢ $ ª $ and since both of these maps are conjugate linear, their composition
is linear.
Here are some of the basic properties of the adjoint.
=
Theorem 10.2 Let and > be finite-dimensional inner product spaces. For
every Á B²= Á > ³ and - ,
1) ²b ³ ~ i i b i
i
2) ² ³ ~ i
3 ) ii ~ and so
i
º#Á $» ~ º#Á $»
i
i
)
4 If =~ > , then ² ³ ~ i
)
³
³
5 If is invertible, then ² c i ~ ² i c
i
)
i
6 If =~ > and ²%³ ´%µ , then ² ³ ~ ² ³ .
s
B
Moreover, if ²= ³ and is a subspace of , then
:
=
i
7 : ) is -invariant if and only if : is -invariant.
)
:
8 ²:Á : ³ reduces if and only if is both -invariant and i -invariant, in
which case
²O³ ~ ² ³O : i : i
Proof. For part 7), let : and ' : and write
i
º'Á » ~ º'Á »
Now, if is -invariant, then º i ' Á » ~ for all : and so i ' : and
:
: i -invariant. Conversely, if : is i -invariant, then º ' Á » ~ is for all
' : and so : ~ :, whence : is -invariant.
The first statement in part 8) follows from part 7) applied to both and : . For
:
the second statement, since is both -invariant and i -invariant, if Á ! , :
:
