View Single Post
jostpuur
jostpuur is offline
#2
Oct6-11, 11:40 AM
P: 1,983
if [itex]A\in \textrm{O}(3,\mathbb{R})[/itex] and [itex]\det(A)=-1[/itex], there exists a [itex]U\in \textrm{SU}(3)[/itex] such that

[tex]
UAU^{\dagger} = \left(\begin{array}{ccc}
-1 & 0 & 0 \\
0 & e^{i\theta} & 0 \\
0 & 0 & e^{-i\theta} \\
\end{array}\right)
[/tex]

with some [itex]\theta\in\mathbb{R}[/itex].

Then there exists a [itex]V\in \textrm{SU}(2)[/itex] such that

[tex]
V\left(\begin{array}{cc}
e^{i\theta} & 0 \\
0 & e^{-i\theta} \\
\end{array}\right)V^{\dagger}
= \left(\begin{array}{cc}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta) \\
\end{array}\right)
[/tex]

So if you define

[tex]
W = \left(\begin{array}{cc}
1 & 0 \\
0 & V \\
\end{array}\right)U
[/tex]

then [itex]WAW^{\dagger}[/itex] will be of such form that reflection and rotation are clearly carried out with respect to the same axis. Only problem is that [itex]W[/itex] doesn't necessarily have only real entries. How to prove that [itex]W[/itex] is necessarily proportional to a real matrix?