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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?

Schach21
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    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. – Malkoun Jan 19 '22 at 04:11
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    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). – Malkoun Jan 19 '22 at 04:17

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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.

Of course, different authors will use different terminology, so it's up to you to determine how the authors of the source in question has chosen to define terms.

Kajelad
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