6
$\begingroup$

It is well known that the $2 \times 2$ rotation matrix is given by, $\left[ \begin{array}{cc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \\ \end{array} \right]$ and that there are $3 \times 3$ rotation matrices to describe rotation in 3-dimensions,

$R_{x}(\theta)=\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos(\theta) & -\sin(\theta) \\ 0 & \sin(\theta) & \cos(\theta) \\ \end{array} \right]$ $R_{y}(\theta)=\left[ \begin{array}{ccc} \cos(\theta) & 0 & \sin(\theta) \\ 0 & 1 & 0 \\ -\sin(\theta) & 0 & \cos(\theta) \\ \end{array} \right]$ $R_{z}(\theta)=\left[ \begin{array}{ccc} \cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1 \\ \end{array} \right]$ That is, there are three elements of the rotation group $SO(3)$, and there is one element of $SO(2)$. In general, I found that for $\phi$ elements of $SO(d)$, where $d$ is the dimension that $\phi = \frac{d(d-1)}{2}$ Yet, how is it that the explicit representation of say $R_{i}(\theta)$ is derived, where $i$ is some arbitrary element of $SO(d)$. Are these things all manually computed or is there a general formula/method for determining what they are? The reason I am asking is that I have a functional in configuration space $T$ that depends on a parameter $R_{j}^{i}$ that is summing over dimensions $i,j$. $R_{j}^{i} \in SO(d)$ is representing an $i \times j$ rotation matrix (note $d=i=j$) and I am having trouble explicitly constructing an $n$-dimensional example since I don't know how to represent an $n \times n$ rotation matrix. Any references to papers, or original responses are welcome.

  • 0
    @RahulNarain: Could you give an explanation so I can accept it as an answer? The wikipedia page is blocked because of the SOPA act awareness thing.2012-01-18

0 Answers 0