Editing Cayley transform (section)

=== Examples ===
In the 2×2 case, we have
:<math>
\begin{bmatrix} 0 & \tan \frac{\theta}{2} \\ -\tan \frac{\theta}{2} & 0 \end{bmatrix}
\leftrightarrow
\begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} .
</math>
The 180° [[rotation matrix]], −''I'', is excluded, though it is the limit as tan&nbsp;<sup>θ</sup>⁄<sub>2</sub> goes to infinity.

In the 3×3 case, we have
:<math>
\begin{bmatrix} 0 & z & -y \\ -z & 0 & x \\ y & -x & 0 \end{bmatrix}
\leftrightarrow
\frac{1}{K}
\begin{bmatrix}
  w^2+x^2-y^2-z^2 & 2 (x y-w z) & 2 (w y+x z) \\
  2 (x y+w z) & w^2-x^2+y^2-z^2 & 2 (y z-w x) \\
  2 (x z-w y) & 2 (w x+y z) & w^2-x^2-y^2+z^2
\end{bmatrix} ,
</math>

where ''K''&nbsp;=&nbsp;''w''<sup>2</sup>&nbsp;+&nbsp;''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>&nbsp;+&nbsp;''z''<sup>2</sup>, and where ''w''&nbsp;=&nbsp;1. This we recognize as the rotation matrix corresponding to [[quaternion]]

:<math> w + \mathbf{i} x + \mathbf{j} y + \mathbf{k} z \,\!</math>

(by a formula Cayley had published the year before), except scaled so that ''w''&nbsp;= 1 instead of the usual scaling so that ''w''<sup>2</sup>&nbsp;+&nbsp;''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>&nbsp;+&nbsp;''z''<sup>2</sup>&nbsp;=&nbsp;1. Thus vector (''x'',''y'',''z'') is the unit axis of rotation scaled by tan&nbsp;<sup>θ</sup>⁄<sub>2</sub>. Again excluded are 180° rotations, which in this case are all ''Q'' which are [[symmetric matrix|symmetric]] (so that ''Q''<sup>T</sup>&nbsp;= ''Q'').