有關旋轉矩的證明

1. 設有旋轉矩陣$R$，證明$R^TR=I$且$det R = \pm1$

證明：假設有兩組正交集$(e_1, e_2, e_3)$與$(e'_1, e'_2, e'_3)$，以及在這兩組正交集下的座標$a = (a_1, a_2, a_3)$與$a' = (a'_1, a'_2, a'_3)$;

根據座標系的定義：

$[e_1, e_2, e_3]\left[\begin{array}{c}a_1\\ a_2\\ a_3\\ \end{array}\right] = [e'_1, e'_2, e'_3]\left[\begin{array}{c}a'_1\\ a'_2\\ a'_3\\ \end{array}\right]$

則

$\left[\begin{array}{c}a_1\\ a_2\\ a_3\\ \end{array} \right] = \left[\begin{array}{ccc} e_1^Te'_1 & e_1^Te'_2 & e_1^Te'_3 \\ e_2^Te'_1 & e_2^Te'_2 & e_2^Te'_3 \\ e_3^Te'_1 & e_3^Te'_2 & e_3^Te'_3 \\ \end{array}\right] \left[\begin{array}{c} a'_1 \\ a'_2 \\ a'_3 \\ \end{array}\right] \doteq Ra' \tag{1}$

其中$R$爲旋轉矩陣，$R^TR$爲：
$R^TR = \left[\begin{array}{ccc} e^T_1e'_1 & e^T_2e'_1 & e^T_3e'_1 \\ e^T_1e'_2 & e^T_2e'_2 & e^T_3e'_2 \\ e^T_1e'_3 & e^T_2e'_3 & e^T_3e'_3 \\ \end{array}\right] \left[\begin{array}{ccc} e_1^Te'_1 & e_1^Te'_2 & e_1^Te'_3 \\ e_2^Te'_1 & e_2^Te'_2 & e_2^Te'_3 \\ e_3^Te'_1 & e_3^Te'_2 & e_3^Te'_3 \\ \end{array}\right] = I \tag{2}$
其中

$(e^T_1e'_1)\times(e^T_1e'_1) + (e^T_2e'_1)\times(e^T_2e'_1)+(e^T_3e'_1)\times(e^T_3e'_1) \\ = (e'_1)^T(e_1e^T_1 + e_2e^T_2+e_3e^T_3)e'_1 = (e_1')^Te'_1 = 1$

由於$1 = det(I)=det(R^TR)=det(R^T)det(R)=(det(R))^2$, 所以$det(R)=\pm1$.

2. 有關四元數的證明

設四元數表示爲$q=(\varepsilon, \eta)$,其中$\varepsilon$爲虛部，$\eta$爲實部。
定義元算$+$和$\oplus$爲:

$q^+=\left[\begin{array}{cc} \eta\mathbf{1}+\varepsilon^\times & \varepsilon \\ -\varepsilon^T & \eta\\ \end{array} \right], q^\oplus= \left[\begin{array}{cc} \eta\mathbf{1}-\varepsilon^\times & \varepsilon \\ -\varepsilon^T & \eta\\ \end{array} \right]$

則：

$\begin{array}{lcl} q_1^+q_2 & = & \left[\begin{array}{cc} \eta_1\mathbf{1}+\varepsilon_1^\times & \varepsilon_1 \\ -\varepsilon_1^T & \eta_1\\ \end{array} \right][\varepsilon_2, \eta_2]^T\\ & = & [\eta_1\varepsilon_2+\varepsilon_1^\times\varepsilon_2+\varepsilon_1\eta_2, -\varepsilon_1^T\varepsilon_2+\eta_1\eta_2]^T \\ & = & q_1q_2 \\ \end{array}$