反对称矩阵的性质

对于向�index.php/tags-41973.html" >�� a = [ a 1 , a 2 , a 3 ] \mathbf a = [a_1,a_2,a_3] a=[a1​,a2​,a3​], 其反对称矩阵为 a ^ = [ a × ] = [ 0 − a 3 a 2 a 3 0 − a 1 − a 2 a 1 0 ] \mathbf a\hat{}= [\mathbf a \times] = \begin{bmatrix}0 & -a_3 & a_2 \\ a_3&0&-a_1 \\ -a_2 &a_1 &0 \end{bmatrix} a^=[a×]= ​0a3​−a2​​−a3​0a1​​a2​−a1​0​ ​

• 对于反对称矩阵 a × a \times a×,存在反交换性 ：

  a × = − [ a × ] T \mathbf a \times = - [\mathbf a \times ]^T a×=−[a×]T

• 叉乘顺序互换，叉乘结果大小不变，方向相反

  （）

  [ a × ] b = − [ b × ] a [\mathbf a \times] \mathbf b= - [\mathbf b \times] \mathbf a [a×]b=−[b×]a

  a T [ b × ] = − b T [ a × ] \mathbf a^T [\mathbf b\times] = - \mathbf b^T [a\times] aT[b×]=−bT[a×]

• 反对称矩阵相加

  [ a × ] + [ b × ] = [ a + b ] × [\mathbf a\times] + [\mathbf b\times] = [\mathbf a+\mathbf b]\times [a×]+[b×]=[a+b]×

• 标量点乘反对称矩阵

  （）

  c ⋅ [ a × ] = [ c a × ] c \cdot [\mathbf a\times]=[c\mathbf a\times] c⋅[a×]=[ca×]

• 向量与自己叉乘等于0向量

  [ a × ] a = 0 [\mathbf a \times] \mathbf a= \mathbf 0 [a×]a=0

• 对于旋转矩阵R，存在：

  [ R a × ] = R [ a × ] R [\mathbf R\mathbf a\times]=\mathbf R[\mathbf a\times] \mathbf R [Ra×]=R[a×]R

  R ( [ a × ] b ) = [ R a × ] ( R b ) \mathbf R([\mathbf a\times] \mathbf b)=[\mathbf R\mathbf a\times] (\mathbf R\mathbf b) R([a×]b)=[Ra×](Rb)

• 混合积

  a ⋅ ( b × c ) = b ⋅ ( c × a ) = c ⋅ ( a × b ) \mathbf a\cdot (\mathbf b\times \mathbf c)= \mathbf b\cdot (\mathbf c\times \mathbf a)=\mathbf c\cdot (\mathbf a\times \mathbf b) a⋅(b×c)=b⋅(c×a)=c⋅(a×b)

• 向量三重积

  a × ( b × c ) = b ( a ⋅ c ) − c ( a ⋅ b ) \mathbf a\times (\mathbf b\times \mathbf c )= \mathbf b(\mathbf a\cdot \mathbf c)-\mathbf c(\mathbf a\cdot \mathbf b) a×(b×c)=b(a⋅c)−c(a⋅b)

  [ a × ] [ b × ] = b a T − a T b I 3 [\mathbf a\times][ \mathbf b\times] = \mathbf b\mathbf a^T-\mathbf a^T\mathbf b\mathbf I_3 [a×][b×]=baT−aTbI3​

• 二次幂公式

  [ a × ] [ a × ] = a a T − ∣ ∣ a ∣ ∣ 2 2 I 3 [\mathbf a\times][ \mathbf a\times] = \mathbf a\mathbf a^T-||\mathbf a||^2_2I_3 [a×][a×]=aaT−∣∣a∣∣22​I3​

• 当 a \mathbf a a不为零向量时

  a × \mathbf a\times a×的秩为2，必有一维零空间，且 a \mathbf a a是其中的一个解

  参考 ：https://zhuanlan.zhihu.com/p/435306687