# Rotation matrices for arbitrary dimensions

I initially asked this question here, and someone suggested this may be a better place to get an answer. I have a question about a rotation matrix, which can be represented in 2 dimensions as: $$R_{2}(\theta)=\begin{bmatrix} cos\theta & sin\theta \\ -sin\theta & cos\theta\end{bmatrix}$$ For some arbitrary angle $\theta$. This can be extended to an arbitrary number of dimensions by adding an identity matrix: $$R_{n}(\theta)=\begin{bmatrix} R_{2}(\theta) & 0 \\ 0 & I_{n-2}\end{bmatrix}$$

My question is: can any rotation in arbitrary dimensions be represented by a sequence of 2-D rotations? (i.e. a product of row/column permuted matrices of the above type?) It doesn't matter if the sequence is unique or not for my purposes.

If this is true (which I think it would be), does adding a translation affect this representation, or does the rotation need to be carried about the origin for the representation to hold?

$$X'=R_{n}^T(X-x_{0})$$

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Yes. This is a consequence of the spectral theorem and is proven, for example, in this blog post. Translations don't affect anything.

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did you make the "stereographic projection" graph in your blog post? What software did you use to do that if so? It looks quite nice! – Qiang Li Feb 21 '11 at 21:09
@Qiang: no, I took it from Wikipedia: en.wikipedia.org/wiki/File:Riemann_sphere1.jpg – Qiaochu Yuan Feb 21 '11 at 21:09