MathDB

SU_2(\mathbb{C})=\left\{\begin{pmatrix} z & w \\ -\bar{w} & \bar{z} \end{pmatrix} : z,w\in\mathbb{C} , z\bar{z}+w\bar{w}=1\right\}

A and B are 2 elements of the above matrix group and have eigenvalues

e^{i\theta_1}

e^{-i\theta_1}

and

e^{i\theta_2}

e^{-i\theta_2}

respectively, where

0\leq\theta_i\leq\pi

. Prove that if AB has eigenvalue

e^{i\theta_3}

, then

\theta_3

satisfies the inequality

|\theta_1-\theta_2|\leq\theta_3\leq \min\{\theta_1+\theta_2 , 2\pi-(\theta_1+\theta_2)\}

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