I want to construct a Rotation matrix in general $n$ dimension along $n$ dimensional vector $\vec{p}$.

First I know how to construct $n=3$ case : consider $n=3$ rotation along $3$ dimensional vectork $\vec{k}$, then

Rotation matrix $R$ is given by \begin{align} \mathbf{R} = \mathbf{I} + \sin(\theta) \mathbf{K} + (1-\cos(\theta)) \mathbf{K} \end{align} where \begin{align} \mathbf {K} =\left[{\begin{array}{ccc}0&-k_{z}&k_{y}\\k_{z}&0&-k_{x}\\-k_{y}&k_{x}&0\end{array}}\right] \end{align}

How about a general $n$ dimensional case?

Is a similar formula exists? then What is it?


I suspect a close-form formula may be hard to come by. However, there is a general prescription for constructing such matrices, though actually doing the construction by hand may not be possible.

Taking as definition that by "rotation" we mean a matrix which leaves the inner product of two vectors $x,y$ invariant: $$ x^{\prime T}y^\prime=x^\prime R^TRy^\prime=x^Ty, $$ where $x^\prime = Rx$ for some rotation matrix $R$, and similarly for $y^\prime$. Such matrices are known as the defining representation of the group $SO(n)$ where the dimension of the vectors is $n$.

There are general results about Lie groups (of which $SO(n)$ is an example) which tell us that there exist so-called generators of the group which when exponentiated give the elements of the group: $$ R(\theta)=e^{\theta^a\omega_a} $$ where $\theta^a$ are the parameters determining the rotation (for example, the Euler angles in $n=3$) and the $\omega_a$ are some collection of yet-to-be-determined matrices.

Now, we took $R^TR=I$ to be the determining relation for our rotations. Let's expand this requirement to first order in $\theta^a$ to find $$ 1=(1+\theta^a\omega_a^T)(1+\theta^a\omega_a)=1+\theta^a(\omega_a^T+\omega_a). $$ Since this must be the case for all (small values of) $\theta^a$, it must be the case that each of the generators $\omega_a$ must satisfy $$ \omega_a=-\omega_a^T, $$ and hence they must all be antisymmetric matrices.

So the result of all this is: in order to generate a rotation in $n$ dimensions, pick some linear combination of antisymmetric $n\times n$ matrices and exponentiate it.

Edit: Let me also add, because I notice you also asked specifically about rotating about a given vector: in order to demand that a given vector $v$ be left invariant by the rotation (so the rotation is about $v$), we are demanding $$ R(\theta)v=v $$ and hence, expanding again to first order in $\theta$, we must have $$ (1+\theta^a\omega_a)v=v. $$ This necessitates $\theta^a\omega_a v=0$. So the collection of rotations which leaves a given vector invariant (rotates about that vector) will be all those rotations formed by an antisymmetric matrix (the linear combination $\theta^a\omega_a$) which contain $v$ in the kernel.


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