Page 257 - Advanced Linear Algebra

Structure Theory for Normal Operators 241

Proof. We leave the proof of part 1) to the reader. For part 2), a
unitary/orthogonal map is injective and since = is finite-dimensional, it is
bijective. Moreover, for a bijective linear map , we have

is an isometry ¯ º #Á $» ~ º#Á $» for all #Á $  =
i
¯ º#Á $» ~ º#Á $» for all #Á $  =
¯ ~ i
¯ i ~ c
¯ is unitary/orthogonal
For part 3), suppose that is unitary/orthogonal and that E ~¸" Á Ã Á " ¹ is an

orthonormal basis for . Then
=

º " Á " »~º" Á " »~ Á

E
and so E is an orthonormal basis for . Conversely, suppose that and E
=
are orthonormal bases for . Then
=
~ º" Á " »
º" Á " » ~ Á
which implies that º #Á $» ~ º#Á$» for all #Á$  = and so is

unitary/orthogonal.

For part 4), if is unitary and #~ # , then
#Á
º#Á #» ~ º #» ~ º #Á #» ~ º#Á #»
and so (( ~ ~ , which implies that (( ~ .
We also have the following theorem concerning unitary and orthogonal)
(
matrices.
Theorem 10.13 Let be an d matrix over -~ d or -~ s .
(
1 The following are equivalent:
)
a ( is unitary/orthogonal.
)
)
b The columns of form an orthonormal set in - .
(
c The rows of form an orthonormal set in - .
)
(
2 If is unitary, then ( ²(³ ~ . If is orthogonal, then det ²(³ ~ f .
)
(det
(
(
(
Proof. The matrix is unitary if and only if ( ( i ~ 0 , which is equivalent to
the rows of ( being orthonormal. Similarly, ( is unitary if and only if
(( ~ 0, which is equivalent to the columns of ( being orthonormal. As for
i
part 2),
i
i
(( ~ 0 ¬ det ²(³det ²( ³ ~ ¬ det ²(³det ²(³ ~
from which the result follows.