本文的閱讀等級：初級
令 
其中 
若 
其中
我們先證明第一式的
欲證明第二式是第一式的反函數，第一式右乘 
當 
![A=\left[\!\!\begin{array}{rc} 0&a\\ -a&0 \end{array}\!\!\right]](https://s0.wp.com/latex.php?latex=A%3D%5Cleft%5B%5C%21%5C%21%5Cbegin%7Barray%7D%7Brc%7D++0%26a%5C%5C++-a%260++%5Cend%7Barray%7D%5C%21%5C%21%5Cright%5D&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle Q=\left[\!\!\begin{array}{rc} 1&a\\ -a&1 \end{array}\!\!\right]^{-1}\left[\!\!\begin{array}{cr} 1&-a\\ a&1 \end{array}\!\!\right]=\left[\!\!\begin{array}{cr} \frac{1-a^2}{1+a^2}&-\frac{2a}{1+a^2}\\ \frac{2a}{1+a^2}&\frac{1-a^2}{1+a^2} \end{array}\!\!\right]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++Q%3D%5Cleft%5B%5C%21%5C%21%5Cbegin%7Barray%7D%7Brc%7D++1%26a%5C%5C++-a%261++%5Cend%7Barray%7D%5C%21%5C%21%5Cright%5D%5E%7B-1%7D%5Cleft%5B%5C%21%5C%21%5Cbegin%7Barray%7D%7Bcr%7D++1%26-a%5C%5C++a%261++%5Cend%7Barray%7D%5C%21%5C%21%5Cright%5D%3D%5Cleft%5B%5C%21%5C%21%5Cbegin%7Barray%7D%7Bcr%7D++%5Cfrac%7B1-a%5E2%7D%7B1%2Ba%5E2%7D%26-%5Cfrac%7B2a%7D%7B1%2Ba%5E2%7D%5C%5C++%5Cfrac%7B2a%7D%7B1%2Ba%5E2%7D%26%5Cfrac%7B1-a%5E2%7D%7B1%2Ba%5E2%7D++%5Cend%7Barray%7D%5C%21%5C%21%5Cright%5D&bg=ffffff&fg=000000&s=0&c=20201002)
使用 
![\displaystyle A=\left[\!\!\begin{array}{cc} 0&\tan\frac{\theta}{2}\\ -\tan\frac{\theta}{2}&0 \end{array}\!\!\right]\rightleftharpoons Q=\left[\!\!\begin{array}{cr} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta \end{array}\!\!\right],~~~\theta\neq \pm\pi](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A%3D%5Cleft%5B%5C%21%5C%21%5Cbegin%7Barray%7D%7Bcc%7D++0%26%5Ctan%5Cfrac%7B%5Ctheta%7D%7B2%7D%5C%5C++-%5Ctan%5Cfrac%7B%5Ctheta%7D%7B2%7D%260++%5Cend%7Barray%7D%5C%21%5C%21%5Cright%5D%5Crightleftharpoons+Q%3D%5Cleft%5B%5C%21%5C%21%5Cbegin%7Barray%7D%7Bcr%7D++%5Ccos%5Ctheta%26-%5Csin%5Ctheta%5C%5C++%5Csin%5Ctheta%26%5Ccos%5Ctheta++%5Cend%7Barray%7D%5C%21%5C%21%5Cright%5D%2C%7E%7E%7E%5Ctheta%5Cneq+%5Cpm%5Cpi&bg=ffffff&fg=000000&s=0&c=20201002)
Cayley 變換可以延伸至複矩陣。若 
考慮這個情況：
在複數平面上，
將虛軸映至單位圓 (扣除
將實軸映至單位圓 (扣除
Cayley transform From Wikipedia
註解
[1] 維基百科：Cayley transform
[2] 假設
若 
上式等號兩邊都是 
線代啟示錄 