How can I show that the associated lie algebra for SO(3) is the set of all 3 dimensional skew-symmetric matrices?
[Math] Lie algebra for SO(3) as a skew symmetric matrix
lie-groupslinear algebra
lie-groupslinear algebra
How can I show that the associated lie algebra for SO(3) is the set of all 3 dimensional skew-symmetric matrices?
Best Answer
$$ \begin{split} \forall A\in SO(3), AA^T=I &\Rightarrow \frac{d}{dt}(AA^T)=0=\dot AA^T+A\dot A^T=\dot AA^T+(\dot AA^T)^T\\ &\Rightarrow (\dot AA^T)^T=-\dot AA^T \end{split} $$ and therefore $\dot AA^T$ is a $3\times 3$ skew symmetric matrix. Therefore, the Lie algebra, which is identified with the tangent space at identity: $$ so(3):=T_eSO(3)=R_{A*}^{-1}T_ASO(3)=\{\dot A(t)A^{-1}(t)|_0|A(0)=A\} $$ is the vector space of all $3\times 3$ skew symmetric matrices ($R_{A*}^{-1}$ is the right pull back).