- #1
LagrangeEuler
- 717
- 22
I'm a little bit confused. Matrices
[tex]\begin{bmatrix}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{bmatrix}[/tex]
##\theta \in [0,2\pi]##
form a group. This is special orthogonal group ##SO(2)##. However it is possible to diagonalize this matrices and get
[tex]\begin{bmatrix}
e^{i\theta} & 0 \\
0 & e^{-i \theta}
\end{bmatrix}=e^{i \theta}\oplus e^{-i\theta}.[/tex]
It looks like that ##e^{i\theta}## is irreducible representation of ##SO(2)##. However in ##e^{i\theta}## we have complex parameter ##i## and this is unitary group ##U(1)##. Where am I making the mistake?
[tex]\begin{bmatrix}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{bmatrix}[/tex]
##\theta \in [0,2\pi]##
form a group. This is special orthogonal group ##SO(2)##. However it is possible to diagonalize this matrices and get
[tex]\begin{bmatrix}
e^{i\theta} & 0 \\
0 & e^{-i \theta}
\end{bmatrix}=e^{i \theta}\oplus e^{-i\theta}.[/tex]
It looks like that ##e^{i\theta}## is irreducible representation of ##SO(2)##. However in ##e^{i\theta}## we have complex parameter ##i## and this is unitary group ##U(1)##. Where am I making the mistake?