# Rotation matrix arbitrary axis?

1. Feb 28, 2013

### Yoran91

Hello everyone.

I'm having some trouble with rotation matrices. I'm given three matrices

$J_1, J_2, J_3$

which form a basis for the set of skew-symmetric matrices ($\mathfrak{so}_3$).
Further, the matrix exponent function is such that

$exp[\alpha J_i]=R_i(\alpha)$,

so taking the exponent of the matrix $J_i$ yields precisely the rotation matrix around the i-th standard basisvector, specified by the angle $\alpha$.

Now I wish to show that any rotation matrix $R(n,\alpha)$ specified by an axis (unit vector) $n$ and angle $\alpha$, can be written as

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

Here, $n_i$ are the components of the unit vector n.
I know I will need the surjectivity of exp for this. Given the surjectivity of exp, how does one show the above is true?