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](adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# General form for 2 x 2 unitary matrices

Loading...

**Physics Forums | Science Articles, Homework Help, Discussion**