What are the properties of rotation axes in N-dimensions?

[mnb96]

Hi,
it is a clear fact that rotations in 3D keep the vectors on the rotation axis unchanged.
In 2D only the zero vector is unchanged.

How can one generalize the concept of rotation axis in N-dimensions?
I've read that for example in 4D one can only rotate around planes.

So are the rotation "hyper-axes" always subspaces of dimensionality n-2?

 

[HallsofIvy]

How do you define "rotation" in n dimensions?

 

[mnb96]

I assumed that "rotations" in n-dimensional Euclidean spaces are represented by the group $$SO(n)$$, since they describe linear transformations which preserve angles and distances.

 

[D H]

The concept of rotating about an axis is special to 3-space. Instead of thinking of rotation in 3-space as being about an axis, think of rotation in 3-space as being parallel to a plane. This way of thinking about rotation does generalize. There is only one plane parallel to the two-space plane, the plane itself. Rotation in 2-space can be described by a single scalar parameter. There are three orthogonal subplanes in 3-space: e.g., the xy, yz, and zx planes. It takes three parameters to describe rotation in 3-space. There are six orthogonal subplanes in 4-space: the three from the xyz 3-subspace plus the wx, yw, and wz subplanes. It takes six scalar parameters to describe rotation in 4-space. This way of thinking makes the two dimension rotation the primitive of rotation in any Euclidean n-space. In general, it takes n(n-1)/2 parameters to describe rotation in n-space, the number of combinations of pairs of axes.

If you want to think of rotation as being about something rather than parallel to a plane, the "about" is a n-2 subspace of the Euclidean n-space. Since 4-2=2, rotation parallel to a plane in 4-space is equivalent to rotation about a plane.

Any rotation in three space can be described in terms of a rotation about a single eigenaxis / parallel to a single eigenplane. This extends to 4-space. Simultaneously rotating parallel to a pair of planes (about a pair of planes) that share a common axis yields a single simple rotation parallel to / about some eigenplane. However, something new happens in four space. Simultaneously rotating about a pair of planes that do not share a common axis (e.g., the xy and wz planes) yields a double rotation. There is no single eigenplane of rotation for such a double rotation.

 

[mnb96]

thanks! your answer was clear!
There is still one issue bugging my mind.

What can we say if we have a $n\times n$ matrix which:
1) belongs to SO(n)
2) has only one real eigenvector

Can we say we are rotating about an axis in n-dimensions or not?

 

[D H]

There is still one issue bugging my mind.

What can we say if we have a $n\times n$ matrix which:
1) belongs to SO(n)
2) has only one real eigenvector

Can we say we are rotating about an axis in n-dimensions or not?

Sure.

Now, can that situation ever arise in 4-space? Think about it for a bit.

 

[mnb96]

I'd need to show that any orthogonal NxN matrix with N even, never has only one real eigenvector but at least two. However I don't know yet how to prove that.

 

[D H]

Think in terms of eigenvalues rather than eigenvectors. What are the implications of the complex conjugate root theorem with regard to the eigenvalues of a real NxN rotation matrix in the case that N is even?

 

[mnb96]

Thanks a lot for the hint!
For some reason I can't remember having studied that important theorem!

I guess the answer is:
We know that the eigenvalues of a matrix NxN are given by the zeroes of a polynomial of degree N.
If we assume the matrix has only one real eigenvalue, it follows from the complex conjugate root theorem that the remaining N-1 roots must be pairs of complex conjugate numbers, but this is not possible because now N-1 is an odd number, so we must have an even number of real eigenvalues.

One last thing:
is it possible matrices in SO(n) with n even to have some real eigenvalue at all? I'd say yes, but I haven't proved it.

 

[JSuarez]

Well, the nxn identity matrix belongs to SO(n), for all n.

 

[D H]

One last thing:
is it possible matrices in SO(n) with n even to have some real eigenvalue at all? I'd say yes, but I haven't proved it.

Sure. Look at any the six primitive rotations in SO(4), for example. One is a rotation about the wz plane,

$$\bmatrix \phantom{-}\cos \theta & \sin \theta & 0 & 0 \\ -\sin\theta & \cos \theta & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \endbmatrix$$

Here 1 is a double eigenvalue with eigenvectors $$\hat z$$ and $$\hat w$$. Any vector with zero x and y components with remain unchanged upon rotation about the wz plane.