# Rotation matrices and exponential map

• Feb 28th 2013, 05:25 AM
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Rotation matrices and exponential map
Hi guys,

I'm having some trouble with rotation matrices. Basically, I want to show that any matrix $R(n \alpha) \in SO(3)$, specified by an angle $\alpha$ and a unit vector $n\in S^2$ can be written as

$R(n \alpha) = exp (\alpha \sum n_i J_i )$,

where $J_i$ is a 'basis' matrix for $\mathfrak{so}(3)$, the space of all skew-symmetric 3x3 matrices. Note that I'm representing a rotation as a vector : the unit vector described the rotation, the length of the vector the amount of degrees

I know the exponential map $exp : \mathfrak{so}(3) \to SO(3)$ is surjective, and that $exp[\alpha J_i]$ coincides with the rotation matrix representing the rotation around the standard basis vector $e_i$ with angle $\alpha$.

I see that surjectivity of exp implies that any R can be written as $R = exp (X)$ for some skew-symmetric matrix X. Given that the $J_i$ span $\mathfrak{so}(3)$, I get

$R= exp (\sum a_i J_i)$.

This is looking like what I want to prove, but I'm kind of stuck here. Can anyone help?