[Mv]_× = M [v]_× M^T的证明
假设 M M M 是一个 3 × 3 3 \times 3 3×3 的旋转矩阵， v = ( v 1 , v 2 , v 3 ) T v = (v_1, v_2, v_3)^T v=(v1,v2,v3)T 是一个三维向量。
下面给出 [ M v ] × = M [ v ] × M T [Mv]_\times = M [v]_\times M^T [Mv]×=M[v]×MT 的推导
1. [ M v ] × [Mv]_\times [Mv]×
计算 M v Mv Mv：
M v = [ m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 ] [ v 1 v 2 v 3 ] = [ m 11 v 1 + m 12 v 2 + m 13 v 3 m 21 v 1 + m 22 v 2 + m 23 v 3 m 31 v 1 + m 32 v 2 + m 33 v 3 ] Mv = \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} = \begin{bmatrix} m_{11}v_1 + m_{12}v_2 + m_{13}v_3 \\ m_{21}v_1 + m_{22}v_2 + m_{23}v_3 \\ m_{31}v_1 + m_{32}v_2 + m_{33}v_3 \end{bmatrix} Mv= m11m21m31m12m22m32m13m23m33 v1v2v3 = m11v1+m12v2+m13v3m21v1+m22v2+m23v3m31v1+m32v2+m33v3
设 M v = ( u 1 , u 2 , u 3 ) T Mv = (u_1, u_2, u_3)^T Mv=(u1,u2,u3)T，则：
[ u ] × = [ 0 − u 3 u 2 u 3 0 − u 1 − u 2 u 1 0 ] [u]_\times = \begin{bmatrix} 0 & -u_3 & u_2 \\ u_3 & 0 & -u_1 \\ -u_2 & u_1 & 0 \end{bmatrix} [u]×= 0u3−u2−u30u1u2−u10
具体形式为：
[ M v ] × = [ 0 − ( m 31 v 1 + m 32 v 2 + m 33 v 3 ) ( m 21 v 1 + m 22 v 2 + m 23 v 3 ) ( m 31 v 1 + m 32 v 2 + m 33 v 3 ) 0 − ( m 11 v 1 + m 12 v 2 + m 13 v 3 ) − ( m 21 v 1 + m 22 v 2 + m 23 v 3 ) ( m 11 v 1 + m 12 v 2 + m 13 v 3 ) 0 ] [Mv]_\times = \begin{bmatrix} 0 & -(m_{31}v_1 + m_{32}v_2 + m_{33}v_3) & (m_{21}v_1 + m_{22}v_2 + m_{23}v_3) \\ (m_{31}v_1 + m_{32}v_2 + m_{33}v_3) & 0 & -(m_{11}v_1 + m_{12}v_2 + m_{13}v_3) \\ -(m_{21}v_1 + m_{22}v_2 + m_{23}v_3) & (m_{11}v_1 + m_{12}v_2 + m_{13}v_3) & 0 \end{bmatrix} [Mv]×= 0(m31v1+m32v2+m33v3)−(m21v1+m22v2+m23v3)−(m31v1+m32v2+m33v3)0(m11v1+m12v2+m13v3)(m21v1+m22v2+m23v3)−(m11v1+m12v2+m13v3)0
2. M [ v ] × M T M [v]_\times M^T M[v]×MT
再看 ( M [v]_\times M^T )：
[ v ] × = [ 0 − v 3 v 2 v 3 0 − v 1 − v 2 v 1 0 ] [v]_\times = \begin{bmatrix} 0 & -v_3 & v_2 \\ v_3 & 0 & -v_1 \\ -v_2 & v_1 & 0 \end{bmatrix} [v]×= 0v3−v2−v30v1v2−v10
计算 ( M [v]_\times M^T )，其中 ( M^T ) 是 ( M ) 的转置：
M [ v ] × M T = [ m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 ] [ 0 − v 3 v 2 v 3 0 − v 1 − v 2 v 1 0 ] [ m 11 m 21 m 31 m 12 m 22 m 32 m 13 m 23 m 33 ] M [v]_\times M^T = \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{bmatrix} \begin{bmatrix} 0 & -v_3 & v_2 \\ v_3 & 0 & -v_1 \\ -v_2 & v_1 & 0 \end{bmatrix} \begin{bmatrix} m_{11} & m_{21} & m_{31} \\ m_{12} & m_{22} & m_{32} \\ m_{13} & m_{23} & m_{33} \end{bmatrix} M[v]×MT= m11m21m31m12m22m32m13m23m33 0v3−v2−v30v1v2−v10 m11m12m13m21m22m23m31m32m33
具体计算每一项：
( 1 , 1 ) : ( − v 3 m 12 + v 2 m 13 ) m 11 + ( v 3 m 11 − v 1 m 13 ) m 21 + ( − v 2 m 11 + v 1 m 12 ) m 31 = 0 (1,1): (-v_3 m_{12} + v_2 m_{13}) m_{11} + (v_3 m_{11} - v_1 m_{13}) m_{21} + (-v_2 m_{11} + v_1 m_{12}) m_{31} = 0 (1,1):(−v3m12+v2m13)m11+(v3m11−v1m13)m21+(−v2m11+v1m12)m31=0
( 1 , 2 ) : ( − v 3 m 12 + v 2 m 13 ) m 12 + ( v 3 m 11 − v 1 m 13 ) m 22 + ( − v 2 m 11 + v 1 m 12 ) m 32 = − ( m 31 v 1 + m 32 v 2 + m 33 v 3 ) (1,2): (-v_3 m_{12} + v_2 m_{13}) m_{12} + (v_3 m_{11} - v_1 m_{13}) m_{22} + (-v_2 m_{11} + v_1 m_{12}) m_{32} = -(m_{31}v_1 + m_{32}v_2 + m_{33}v_3) (1,2):(−v3m12+v2m13)m12+(v3m11−v1m13)m22+(−v2m11+v1m12)m32=−(m31v1+m32v2+m33v3)
( 1 , 3 ) : ( − v 3 m 12 + v 2 m 13 ) m 13 + ( v 3 m 11 − v 1 m 13 ) m 23 + ( − v 2 m 11 + v 1 m 12 ) m 33 = ( m 21 v 1 + m 22 v 2 + m 23 v 3 ) (1,3): (-v_3 m_{12} + v_2 m_{13}) m_{13} + (v_3 m_{11} - v_1 m_{13}) m_{23} + (-v_2 m_{11} + v_1 m_{12}) m_{33} = (m_{21}v_1 + m_{22}v_2 + m_{23}v_3) (1,3):(−v3m12+v2m13)m13+(v3m11−v1m13)m23+(−v2m11+v1m12)m33=(m21v1+m22v2+m23v3)
( 2 , 1 ) : ( v 3 m 11 − v 1 m 13 ) m 11 + ( − v 3 m 12 + v 2 m 13 ) m 21 + ( v 3 m 11 − v 1 m 13 ) m 31 = ( m 31 v 1 + m 32 v 2 + m 33 v 3 ) (2,1): (v_3 m_{11} - v_1 m_{13}) m_{11} + (-v_3 m_{12} + v_2 m_{13}) m_{21} + (v_3 m_{11} - v_1 m_{13}) m_{31} = (m_{31}v_1 + m_{32}v_2 + m_{33}v_3) (2,1):(v3m11−v1m13)m11+(−v3m12+v2m13)m21+(v3m11−v1m13)m31=(m31v1+m32v2+m33v3)
( 2 , 2 ) : ( v 3 m 11 − v 1 m 13 ) m 12 + ( − v 3 m 12 + v 2 m 13 ) m 22 + ( v 3 m 11 − v 1 m 13 ) m 32 = 0 (2,2): (v_3 m_{11} - v_1 m_{13}) m_{12} + (-v_3 m_{12} + v_2 m_{13}) m_{22} + (v_3 m_{11} - v_1 m_{13}) m_{32} = 0 (2,2):(v3m11−v1m13)m12+(−v3m12+v2m13)m22+(v3m11−v1m13)m32=0
( 2 , 3 ) : ( v 3 m 11 − v 1 m 13 ) m 13 + ( − v 3 m 12 + v 2 m 13 ) m 23 + ( v 3 m 11 − v 1 m 13 ) m 33 = − ( m 11 v 1 + m 12 v 2 + m 13 v 3 ) (2,3): (v_3 m_{11} - v_1 m_{13}) m_{13} + (-v_3 m_{12} + v_2 m_{13}) m_{23} + (v_3 m_{11} - v_1 m_{13}) m_{33} = -(m_{11}v_1 + m_{12}v_2 + m_{13}v_3) (2,3):(v3m11−v1m13)m13+(−v3m12+v2m13)m23+(v3m11−v1m13)m33=−(m11v1+m12v2+m13v3)
( 3 , 1 ) : ( − v 2 m 11 + v 1 m 12 ) m 11 + ( v 3 m 11 − v 1 m 13 ) m 21 + ( − v 2 m 11 + v 1 m 12 ) m 31 = − ( m 21 v 1 + m 22 v 2 + m 23 v 3 ) (3,1): (-v_2 m_{11} + v_1 m_{12}) m_{11} + (v_3 m_{11} - v_1 m_{13}) m_{21} + (-v_2 m_{11} + v_1 m_{12}) m_{31} = -(m_{21}v_1 + m_{22}v_2 + m_{23}v_3) (3,1):(−v2m11+v1m12)m11+(v3m11−v1m13)m21+(−v2m11+v1m12)m31=−(m21v1+m22v2+m23v3)
( 3 , 2 ) : ( − v 2 m 11 + v 1 m 12 ) m 12 + ( v 3 m 11 − v 1 m 13 ) m 22 + ( − v 2 m 11 + v 1 m 12 ) m 32 = ( m 11 v 1 + m 12 v 2 + m 13 v 3 ) (3,2): (-v_2 m_{11} + v_1 m_{12}) m_{12} + (v_3 m_{11} - v_1 m_{13}) m_{22} + (-v_2 m_{11} + v_1 m_{12}) m_{32} = (m_{11}v_1 + m_{12}v_2 + m_{13}v_3) (3,2):(−v2m11+v1m12)m12+(v3m11−v1m13)m22+(−v2m11+v1m12)m32=(m11v1+m12v2+m13v3)
( 3 , 3 ) : ( − v 2 m 11 + v 1 m 12 ) m 13 + ( v 3 m 11 − v 1 m 13 ) m 23 + ( − v 2 m 11 + v 1 m 12 ) m 33 = 0 (3,3): (-v_2 m_{11} + v_1 m_{12}) m_{13} + (v_3 m_{11} - v_1 m_{13}) m_{23} + (-v_2 m_{11} + v_1 m_{12}) m_{33} = 0 (3,3):(−v2m11+v1m12)m13+(v3m11−v1m13)m23+(−v2m11+v1m12)m33=0
整理后的结果为：
M [ v ] × M T = [ 0 − ( m 31 v 1 + m 32 v 2 + m 33 v 3 ) ( m 21 v 1 + m 22 v 2 + m 23 v 3 ) ( m 31 v 1 + m 32 v 2 + m 33 v 3 ) 0 − ( m 11 v 1 + m 12 v 2 + m 13 v 3 ) − ( m 21 v 1 + m 22 v 2 + m 23 v 3 ) ( m 11 v 1 + m 12 v 2 + m 13 v 3 ) 0 ] M [v]_\times M^T = \begin{bmatrix} 0 & -(m_{31}v_1 + m_{32}v_2 + m_{33}v_3) & (m_{21}v_1 + m_{22}v_2 + m_{23}v_3) \\ (m_{31}v_1 + m_{32}v_2 + m_{33}v_3) & 0 & -(m_{11}v_1 + m_{12}v_2 + m_{13}v_3) \\ -(m_{21}v_1 + m_{22}v_2 + m_{23}v_3) & (m_{11}v_1 + m_{12}v_2 + m_{13}v_3) & 0 \end{bmatrix} M[v]×MT= 0(m31v1+m32v2+m33v3)−(m21v1+m22v2+m23v3)−(m31v1+m32v2+m33v3)0(m11v1+m12v2+m13v3)(m21v1+m22v2+m23v3)−(m11v1+m12v2+m13v3)0
总结
经过计算，可以看到 [ M v ] × 和 M [ v ] × M T [Mv]_\times 和 M [v]_\times M^T [Mv]×和M[v]×MT 的形式确实是一致的，因此等式 [ M v ] × = M [ v ] × M T [Mv]_\times = M [v]_\times M^T [Mv]×=M[v]×MT 是成立的。