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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?

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

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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 $b = e^{i \alpha_2} \sin \phi$.

Another context where this kind of redundancy in parameterization arises is in the definition of spherical coordinates.

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  • $\begingroup$ Thank you. It helped : ) $\endgroup$ Commented Aug 22, 2022 at 16:34

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