５．证明Ｒp^R' = (Rp)^
b = p x c = p^ c, 前一个 x 为叉乘, 后一个 ^ 为取反对称矩阵.
由于Rb,Rc,Rp是做空间旋转,那么有Rb = Rp x Rc.
Rb = R(p x c) = Rp x Rc = (Rp)^ Rc,
b = R'(Rp)^ Rc. " ' " 为转置
b = p^ c = R'(Rp)^ R c
p^ = R'(Rp)^ R
Rp^ R' = (Rp)^.
6.1)证明 R exp(p^)R' = exp((Rp)^)
由5中证明可知:
exp((Rp)^) = exp(Rp^R')
由级数以及R'R = I可知
exp(Rp^R') = R exp(p^) R'
则由 exp((Rp)^) = R exp(p^) R'
2)证明 T exp($^) T` = exp((Ad(T)$)^), " ` " 为求逆
由exp(x)的级数可知: T exp($^)T` = exp(T$^T`)
所以只需证: T$^T` = (Ad(T)$)^