From <a href=http://math.ucr.edu/home/baez/lie/node6.html>here</a>:
SU(2) is isomorphic to the group of quaternions of norm 1!
<table><tr><td>U = <td><table><tr><td>a+id<td>c+ib<tr><td>−c+ib<td>a−id</table><td>= a<td><table><tr><td>1<td>0<tr><td>0<td>1</table>
<td>+ b<td> <table><tr><td>0<td>i<tr><td>i<td>0</table>
<td>+ c<td> <table><tr><td>0<td>1<tr><td>−1<td>0</table>
<td>+ d<td> <table><tr><td>i<td>0<tr><td>0<td>−i</table>
</table>
= a<b>1</b> + b<b>i</b> + c<b>j</b> + d<b>k</b>.
<br>a<sup>2</sup> + b<sup>2</sup> + c<sup>2</sup> + d<sup>2</sup> = 1.
<br>U* is the conjugate transpose of U.
<p><table><tr><td>U is <b>unitary</b> iff<td>U*U =<td><table><tr><td>1<td>0<tr><td>0<td>1</table></table>
<a href=tools>This code</a> (in <a href=../../../code/Scheme/reposIntro>this framework</a>) confirms that these U’s are indeed unitary.
<p>For every 2 by 2 complex matrix A, there is a unitary U and complex e and f such that:
<table><tr><td>U*AU = <td><table><tr><td>e<td>0<tr><td>0<td>f</table></table>
<hr>Play:
<p>above
<br>front<ruby>WWW<rp>(<rt>World Wide Web<rp>)</ruby>back<ruby>x<rp>(<rt>⋀<rp>)</ruby>
</br>below ⋀⋏⌃