# Confusion about connection between orthogonal matrices and rotation in higher dimensions

I found this question which discusses that all orthogonal matrices are rotations/reflections, since the map $$X\rightarrow AX$$ preserves the scalar product, with $$A$$ an orthogonal matrix (see proof in the question mentioned).

Before I had found a resource that said that a $$4$$ (or higher) dimensional rotation matrix should be of the form: $$\begin{bmatrix} \cos(\theta) & -\sin(\theta) & 0 & 0 \\ \sin(\theta) & \cos(\theta) & 0 & 0 \\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}$$

Basically, it has the usual $$\cos,\sin$$ structure of the $$2$$ and $$3$$D rotations, with the off-diagonal elements being $$0$$ and the diagonal elements being $$1$$.

This definition of rotation considers that operation as going from one axis to another, i.e. rotation in $$2$$D is around a point, where the $$x$$-axis goes towards the $$y$$-axis, rotation in $$3$$D is around a line and this time there are different options: $$x$$-axis goes towards $$y$$-axis or $$z$$-axis, $$y$$-axis towards $$z$$-axis, etc. This suggests that rotation in $$4$$D happens over a plane, etc. This constructions imposes a certain structure on rotation matrices.

This basically implies that there orthogonal matrices that are not rotation in the sense I just defined.

So, is it correct to say/more precise to say that all orthogonal matrices are rotations/reflections in that the map $$X\rightarrow AX$$ preserves the scalar product. But if one considers the definition I used of rotation, then not all orthogonal matrices are rotation matrices? Or is there a way to bridge these two seemingly different definitions?

I vaguely remember when I was reading about this that they mentioned through a change of basis you reach any orthogonal matrix from a rotation matrix with the structure I mentioned above. Is this true?

• In dimension 4, there are other rotations than the one mentioned. For example, you can consider the direct sum of two planar rotations of say different angles, $\theta_1$ and $\theta_2$. But any rotation in Euclidean 4-space can be written as a finite product of "basic" rotations which are of the kind mentioned. I hope this helps. Jan 19, 2022 at 4:11
• I think that in dimension 4, any rotation can be written as a direct sum of 2 planar rotations, after changing basis to a new orthonormal basis if needed. In general dimension $d$, if $d = 2m$, then I think you can write a rotation as a direct sum of $m$ planar rotations (after changing to a different orthonormal basis if needed). If $d = 2m+1$, then I think you can write a rotation as a direct sum of $m$ planar rotations and a $1$ by $1$ identity "matrix", generalizing the case $d = 3$ (after changing to a different orthonormal basis, if needed). Jan 19, 2022 at 4:17

The standard terminology is that rotation can refer to any element of $$SO(n)$$.
(Alternately, in some contexts all elements of $$O(n)$$ are called rotations and elements of $$SO(n)$$ are called proper rotations.)
The matrices you describe are most often referred to as simple rotations or planar rotations. In dimension $$\le 3$$ all rotations are simple, but in higher dimensions, rotations can act nontrivially on multiple orthogonal planes; such rotations are not simple.