Eigenvectors of linear transformations: Reflections vs Rotations

Question

I'm curious why reflections can have real eigenvectors/eigenvalues whereas rotations always have imaginary numbers. The two linear transformations seem similar to me in spirit so this difference is striking to me.

Answer 1

Let $A$ be a matrix that reflect vectors about an arbitrary line $\mathscr{L}$ with angle $\theta$.
Let $s=\sin(\theta)$ and $c=\cos(\theta)$ for convenience. Then we write $A$ as shown $$A=\begin{bmatrix}c&-s\\s&c\end{bmatrix}\begin{bmatrix}1&0\\0&-1\end{bmatrix}\begin{bmatrix}c&s\\-s&c\end{bmatrix}=\begin{bmatrix}c^2-s^2&2sc\\2sc&s^2-c^2\end{bmatrix}$$ This matrix will reflect any vector about the line $\mathscr{L}$, and the eigenvalues of $A$ are always $\pm 1$.

Let $R$ be a matrix that rotates a vector by an arbitrary angle $\theta$: $$R=\begin{bmatrix}c&-s\\s&c\end{bmatrix}$$ We get eigenvalues $\lambda=c\pm is$ which are complex for all $\theta\ne n\pi$.

Now, at first glance this difference seems weird considering that any vector that gets reflected about an arbitrary line $\mathscr{L}$ can also be rotated to end up at the same configuration. That is, for any reflection $A\mathbf{v}$ we can always find a rotation $R\mathbf{v}$ such that $A\mathbf{v}=R\mathbf{v}$. However, this will hold only for that particular vector $\mathbf{v}$.
To dig deeper, lets look at the eigenvectors.

The eigenvector for $\lambda_1=1$ and $\lambda_2=-1$ for the reflection matrix $A$ can be written $$\mathbf{v}_1=\begin{bmatrix}\cos(\theta)\\ \sin(\theta)\end{bmatrix},\hspace{1cm}\mathbf{v}_2=\begin{bmatrix}-\sin(\theta)\\\cos(\theta)\end{bmatrix}$$ The eigenvectors for $\lambda_1=(c+is)$ and $\lambda_2=(c-is)$ for the rotation matrix $R$ can be written $$\mathbf{v}_1=\frac{1}{\sqrt{2}}\begin{bmatrix}-1\\i\end{bmatrix}, \hspace{1cm}\mathbf{v}_2=\frac{1}{\sqrt{2}}\begin{bmatrix}1\\i\end{bmatrix}$$ To conclude, we see that the reflection matrix has an eigenvector along the line of reflection with positive unity eigenvalue and an eigenvector perpendicular to the line of reflection with negative unity eigenvalue.
The rotation matrix rotate the eigenvectors by the angle $\theta$. To enable for vector-rotation by scalar multiplication, i.e. $\lambda\mathbf{v}=R\mathbf{v}$, both the eigenvector and eigenvalue must be complex.