In 2D a square matrix is a rotation of the plane, can $n \times n$ matrices be thought of as rotations in $\mathbb{R}^n$?

Question

In 2D a square matrix is a rotation of the plane, can $n \times n$ matrices be thought of as rotations in $\mathbb{R}^n$?

I just want to clarify some of my intuitions about (square) matrices...

Answer 1

Not all matrices are rotations. Rotation matrices in 2D are of the form $\begin{bmatrix} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$.

For example, the matrix $2I$ is not a rotation -- it just scales up the vector to twice its length. And a reflection isn't a rotation.

One can define rotations as matrices which are orthogonal (their transpose is their inverse) and have determinant 1.

Answer 2

A square matrix is not always a rotation of the plane : take $$s=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ which is a symetry. The rotations are only $$R=\Big\{\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}\mid\theta\in[-\pi,\pi]\Big\}.$$