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I've been working through "Groups and Symmetry" (Armstrong) and came across this problem in chapter 9 which I can't figure out. Any hints/help would be greatly appreciated!

Show that every $2\times2$ unitary matrix has the form

$ \left(\begin{array}{c c} w & z \\ -e^{i \theta} z^{*} & e^{i \theta} w^{*} \end{array}\right) $

for some $\theta$ real, and $w$, $z$ complex. (A matrix is said to be unitary if it is invertible with its adjoint as the inverse. The symbol "*" denotes complex conjugate.)

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One more restriction is necessary, that $ww^* + zz^* =1$ –  adam W Jan 4 '13 at 4:17

1 Answer 1

Start with the facts you know, i.e. that you have a 2-by-2 complex matrix $\begin{pmatrix}w&z\\c&d\end{pmatrix}$, such that when you multiply it by its adjoint $\begin{pmatrix}w^*&c^*\\z^*&d^*\end{pmatrix}$ you get $\begin{pmatrix}1&0\\0&1\end{pmatrix}$. That means you have $ww^*+zz^* = 1$, $cc^*+dd^* = 1$ and $cw^*+dz^*=0$. Don't forget the other way, so you get $ww^*+cc^* = 1$, $zz^*+dd^* = 1$ and $w^*z+c^*d=0$. With these equations you should notice something immediately about both $cc^*$ and $dd^*$. You can work from there.

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