- #1

Ken Gallock

- 30

- 0

There is the ##2\times 2## matrix ##B##

$$B=

\left[

\begin{array}{cc}

B_{11} &B_{12} \\

B_{21}&B_{22}

\end{array}

\right],~B_{ij}\in \mathbb{C}

$$

with property

$$\vert B_{11}\vert^2 + \vert B_{12}\vert^2=1,$$

$$\vert B_{21}\vert^2 + \vert B_{22}\vert^2=1,$$

$$B_{11}B_{21}^{\ast}+B_{12}B_{22}^{\ast}=0.$$

According to one of the texts, it is said that this matrix can be decomposed like

$$B=e^{i\frac{\Lambda}{2}}

\left[

\begin{array}{cc}

e^{i\frac{\Phi}{2}} & 0 \\

0 & e^{-i\frac{\Phi}{2}}

\end{array}

\right]

\left[

\begin{array}{cc}

\cos (\Theta/2) & \sin (\Theta/2) \\

-\sin (\Theta/2) & \cos (\Theta/2)

\end{array}

\right]

\left[

\begin{array}{cc}

e^{i\frac{\Psi}{2}} & 0 \\

0 & e^{-i\frac{\Psi}{2}}

\end{array}

\right]

$$

$$\Lambda, \Phi, \Theta, \Psi \in \mathbb{R}$$.

I don't know what kind of decomposition this is.

Could someone tell me the name of this decomposition?

Thanks.