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?
Best Answer
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.