# Possible value of angles in parametrization of $2\times 2$ unitary matrices

I am doing parametrization of $$2\times 2$$ matrices from book 'The Unitary and Rotation Groups' by F.D. Murnaghan. Writing the unitary matrix as $$U = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$$ and applying the condition $$U^\dagger U = I_{2\times 2}$$ we get equations like $$a^*a + b^*b = 1$$ $$c^*c + d^*d = 1$$ $$a^*c + b^*d = 0$$ where $$a, b, c, d \in \mathbb{C}$$.

So, we can write for the first equation: $$a = e^{\iota \alpha_1}\cos\phi$$ and $$b = e^{\iota \alpha_2}\sin \phi$$ Then what are the bounds on the values of $$\alpha_1, \alpha_2 \ \& \ \phi ?$$ I can understand that $$\phi$$ can vary from $$-\pi$$ to $$\pi$$, but so should $$\alpha_1$$ and $$\alpha_2$$. But the book says that the latter angles can have values between $$-\pi/2$$ to $$\pi/2$$ and terms them as latitude angles, while $$\phi$$ as longitude angle. Why is there a constraint on the value of $$\alpha$$'s?

• Please do not rely on pictures of text. Commented Aug 22, 2022 at 14:12
• Commented Aug 22, 2022 at 14:13
• @Shaun I have formatted it now with MathJax. Thanks for the tutorial. Waiting for an answer now : ) Commented Aug 22, 2022 at 15:34
• You're welcome :) Commented Aug 22, 2022 at 15:37
• @Dan That is just the third equation after taking a complex conjugate. So, it doesn't provide anything new. Commented Aug 22, 2022 at 16:35

The problem with taking $$a = e^{i \alpha_1} \cos \phi$$ where $$\alpha_1$$ and $$\phi$$ are both taken from $$-\pi$$ to $$\pi$$ is that it's a redundant parameterization; that is, the same $$a$$ corresponds to multiple possible combinations of $$\phi$$ and $$\alpha_1$$. For example, $$a = -1$$ can be reached either by using $$\alpha_1 = 0$$ and $$\phi = \pi$$ or $$\alpha_1 = \pi$$ and $$\phi = 0$$.
In order to avoid this redundancy, we need to either restrict the possible values of $$\phi$$ or the possible values of $$\alpha_1$$. In this case, the author has decided to restrict the possible values of $$\alpha_1$$. A similar situation arises with $$b = e^{i \alpha_2} \sin \phi$$.