I want to be able to write: "Suppose $A$ is a $3 \times 3$ orthogonal matrix. Then $A$ is of one of the following forms: ..." I need some parametrization of a general $3 \times 3$ orthogonal matrix.
Wikipedia gives the following parametrization on its page on orthogonal matrices:
$$\small\begin{bmatrix} \cos (\alpha)\cos (\gamma)-\sin (\alpha)\sin (\beta)\sin (\gamma) & -\sin (\alpha)\cos (\beta) & -\cos (\alpha)\sin (\gamma)-\sin (\alpha)\sin (\beta)\cos (\gamma) \\ \cos (\alpha)\sin (\beta)\sin (\gamma)+\sin (\alpha)\cos (\gamma) & \cos (\alpha)\cos (\beta) & \cos (\alpha)\sin (\beta )\cos (\gamma )-\sin (\alpha )\sin (\gamma) \\ \cos (\beta)\sin (\gamma) & -\sin (\beta) & \cos (\beta)\cos (\gamma) \end{bmatrix}$$
The article is very confusingly written right now. This matrix is given without much information. It's parametrized by three angles $\alpha, \beta, \gamma$, and I've checked its determinant to be $1$ using Mathematica, so I assume that this parametrizes the orthogonal $3 \times 3$ matrices with determinant $1$, i.e., all rotations about the origin. However, it also says:
but [...] the non-rotational matrices can be more complicated than reflections.
mentioning something about rotoinversions, and
However, we have elementary building blocks for permutations, reflections, and rotations that apply in general.
without further explanation of what on earth that means.
Am I correct in assuming that all orthogonal matrices are either parametrized by the above matrix or its additive inverse? In other words, would adding a $\pm$ sign in front of the above parametrization for $\mathrm{SO}(3)$ give me one for $\mathrm{O}(3)$?