- #1
jimmycricket
- 116
- 2
I'm trying to show that any unitary matrix may be written in the form [itex]\begin{pmatrix}e^{i\alpha_1}\cos{\theta} & -e^{i\alpha_2}\sin{\theta}\\ e^{i\alpha_3}\sin{\theta} & e^{i\alpha_4}\cos{\theta}\end{pmatrix}[/itex]
Writing the general form of a unitary matrix as
[itex]U=\begin{pmatrix} u_{11} & u_{12}\\ u_{21} & u_{22}\end{pmatrix}[/itex]
gives
[itex]U^{\dagger}U=
\begin{pmatrix}u_{11}^* & u_{21}^*\\u_{12}^* & u_{22}^*\end{pmatrix}\begin{pmatrix} u_{11} & u_{12}\\ u_{21} & u_{22}\end{pmatrix}=\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}[/itex]
[itex]\Longrightarrow |u_{11}|^2+|u_{21}|^2=1 \:\:,\:\: |u_{12}|^2 + |u_{22}|^2=1\\
\Longrightarrow |u_{11}|=\cos(\theta) \:\:,\:\: |u_{21}|=\sin(\theta) \:\:,\:\: |u_{12}|=\cos(\varphi) \:\:,\:\: |u_{22}|=\sin(\varphi)[/itex]
for some [itex]\theta , \varphi[/itex]
I'm not really sure where to go from here.
Writing the general form of a unitary matrix as
[itex]U=\begin{pmatrix} u_{11} & u_{12}\\ u_{21} & u_{22}\end{pmatrix}[/itex]
gives
[itex]U^{\dagger}U=
\begin{pmatrix}u_{11}^* & u_{21}^*\\u_{12}^* & u_{22}^*\end{pmatrix}\begin{pmatrix} u_{11} & u_{12}\\ u_{21} & u_{22}\end{pmatrix}=\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}[/itex]
[itex]\Longrightarrow |u_{11}|^2+|u_{21}|^2=1 \:\:,\:\: |u_{12}|^2 + |u_{22}|^2=1\\
\Longrightarrow |u_{11}|=\cos(\theta) \:\:,\:\: |u_{21}|=\sin(\theta) \:\:,\:\: |u_{12}|=\cos(\varphi) \:\:,\:\: |u_{22}|=\sin(\varphi)[/itex]
for some [itex]\theta , \varphi[/itex]
I'm not really sure where to go from here.