例三：The rotation group SO(3), its Lie algebra so(3), and the vector space R^3

Example 3:The rotation group SO(3), its Lie algebra so(3), and the vector space R^3

In the rotation group SO(3), of 3×3 rotation matrices R, we have the orthogonality condition $R^{t}R$ = I. The tangent space may be found by taking the time derivative of this constraint, that is $R^t\dot{R}+{\dot{R}}^{t}R=0$, which we rearrange as
$$R^t\dot{R}=-(R^t\dot{R}{)}^t$$
This expression reveals that $R^t\dot{R}$ is a skew-symmetric matrix (the negative of its transpose). Skew-symmetric matrices are often noted $[\omega {]}_{\times }$ and have the form

$$[\omega {]}_{\times }=\left[\begin{array}{ccc}0& -{\omega }_x& {\omega }_y\\ {\omega }_x& 0& -{\omega }_x\\ -{\omega }_y& {\omega }_x& 0\end{array}\right]$$
This gives $R^t\dot{R}=[\omega {]}_{\times }$. When R = I we have
$$\dot{R}=[\omega {]}_{\times },$$
that is, $[\omega {]}_{\times }$ is in the Lie algebra of SO(3), which we name so(3). Since $[\omega {]}_{\times }$ ∈ so(3) has 3 DoF, the
dimension of SO(3) is m = 3. The Lie algebra is a vector space whose elements can be decomposed into
$$[\omega {]}_{\times }={\omega }_x{E}_x+{\omega }_y{E}_y+{\omega }_z{E}_z$$
$with{E}_x=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& -1\\ 0& 1& 0\end{array}\right],E_y=\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ -1& 0& 0\end{array}\right]，E_z=\left[\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right]$
the generators of so(3), and where ω = (ωx, ωy, ωz) ∈ R3 is the vector of angular velocities. The one-to-one linear relation above allows us to identify so(3) with R3 —we write $so(3)\simeq R^3$. We pass from so(3) to R3 and viceversa using the linear operators hat and vee,

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在3×3旋转矩阵R的旋转群SO(3)中，我们有正交条件$R^{t}R=I$。切线空间可以通过对该约束的时间导数求出，即 $R^t\dot{R}+{\dot{R}}^{t}R=0$，我们将其重新排列为
$$R^t\dot{R}=-(R^t\dot{R}{)}^t$$
即$R^t\dot{R}$ 是一个偏对称矩阵（其转置的负数）。 斜对称矩阵通常记为 $[\omega {]}_{\times }$并具有以下形式
$$[\omega {]}_{\times }=\left[\begin{array}{ccc}0& -{\omega }_x& {\omega }_y\\ {\omega }_x& 0& -{\omega }_x\\ -{\omega }_y& {\omega }_x& 0\end{array}\right]$$
也就是 $R^t\dot{R}=[\omega {]}_{\times }$. 当 R = I 时
$$\dot{R}=[\omega {]}_{\times },$$

也就是说，$[\omega {]}_{\times }$ 在 SO(3) 的李代数中，我们将其命名为 so(3)。 由于 $[\omega {]}_{\times }$ ∈ so(3) 有 3 个自由度，所以SO(3) 的维数是 m = 3。李代数是一个向量空间，其元素可以分解为
$$[\omega {]}_{\times }={\omega }_x{E}_x+{\omega }_y{E}_y+{\omega }_z{E}_z$$
$其中E_x=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& -1\\ 0& 1& 0\end{array}\right],E_y=\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ -1& 0& 0\end{array}\right]，E_z=\left[\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right]是so(3)的基$
角速度的向量ω = (ωx, ωy, ωz) ∈ R3。存在so(3) 与 R3间的一一线性关系，即$so(3)\simeq R^3$. 使用线性算子hat 和 vee再两个空间转换。
$$\begin{gathered} Hat:R3\to so(3);\omega \to {\omega }^{\wedge }=[\omega {]}_{\times } \\ Vee:so(3)\to R3;[\omega {]}_{\times }\to [\omega {]}_{\times }^{\vee }=\omega . \end{gathered}$$