Any reference(a book) that defines the $n$-dimensional rotation matrix?

Question

I want to refer to a mathematics book that explains the n-dimensional rotation matrix or rotation transformation.

Wikipedia concentrates most on 2D or 3D. There are things that one can say definition here and there, but I think it is not a good idea to use the definition there. Actually they don't seem to be definitions.

Strang's "Linear Algebra", Barret O'neill's "Elementary Differential Geometry" deal only with 2D or 3D cases. I think physicist are more interested in the general case, due to the theory of relativity. I found one explanation in "Geometrical Methods of Mathematical Physics" by Bernard Schutz. But I think it doesn't define the rotation matrix.

Artin's "Geometric Alegebra" defines the rotation group as an isometry $\sigma:V\to V$ such that $\det\sigma=1$. But the language there is so abstract that I can't catch any of them.

Can anyone give a reference that defines rotation transformation on $\mathbb R^n$ and state as a property that $A$ is a rotation matrix if and only if $A\in SO(n)$?

This is the end of the question and the below is what I wanted to do. I wanted to prove that if $A\in SO(n)$, then $A$ is a rotation about a line through the origin in $\mathbb R^n$. So I need to define the rotation transformation(or matrix) in $n$ dimensional Euclidean space.

Answer 1

The definition Artin gives is probably the most transparent and standard:

1. It is a linear isometry, meaning that it preserves distances and lengths as given by a bilinear form (You may want to restrict yourself to the dot product in $\mathbb R^n$).
2. It has determinant $1$, meaning that it doesn't change the orientation of the space by reflecting it ($\mathbb R$-linear isometries must have determinant $\pm 1$, so we are excluding half with this condition.)

I think you should spend time trying to understand this rather than discarding it as "too abstract." It's really quite concrete.

But I think it doesn't define the rotation matrix.

To split hairs for a second, the most important nature of a rotation is that it is a transformation, not just a matrix. A matrix is just a particular way to represent a transformation. That's why the definition above emphasizes the qualities that make it a rotation: it does not mess with distances, and it does not mess with orientation. (Out of infinitely many choices of bases it can have infinitely many different matrix representations.)

I wanted to prove that if $A\in SO(n)$, then $A$ is a rotation about a line through the origin in $\mathbb R^n$.

That isn't even true in $\mathbb R^2$, as I'm sure you can see.

It happens to be true for rotations in $\mathbb R^n$ for odd $n$ due to the fundamental theorem of algebra, because it says each rotation (like in Artin's definition) has an eigenvector. But even then I doubt it's what you wanted: some rotations in $\mathbb R^n$ preserve much more than just one line through the origin.

And for even $n$'s, you still might not preserve any line: take, for example

$\begin{bmatrix}0&-1&0&0 \\ 1&0&0&0 \\ 0&0&0&-1\\ 0&0&1&0\end{bmatrix} $ as an $\mathbb R$ linear transformation.