find the eigenvalues and eigenvectors of the reflection matrix.

Question

Let $R_{\theta} = \pmatrix{\cos \theta & \sin \theta \\ \sin \theta & -\cos \theta}$.

So I was able to find that the eigenvalues are $1$ and $-1$ using $\det(R_{\theta} - \lambda I)$.

I also know how to find the eigenvectors if I use the reflection matrix $\pmatrix{0 & 1 \\ 1 & 0}$, so I know the two eigenvectors are

$$\pmatrix{1 \\ 1}, \pmatrix{1 \\ -1}.$$

My question is: how do I find the eigenvectors using the cosine and sine matrix $R_{\theta}$ instead of the easier matrix? Can I do things (I feel like the answer is no, but googling didn't tell me) like square a row in my attempts to reduce the matrix down? Does that make sense what I am asking? If not, I am happy to try to clarify.

Answer 1

Lets take an example using one of the eigenvalues.

For $\lambda = 1$, we want to find the RREF of $[A - \lambda I]v_1 = [A - I]v_1 = 0$. This gives

$$\begin{pmatrix} \cos \theta -1 & \sin \theta \\ \sin \theta & -(\cos \theta +1)\end{pmatrix}v_1 = 0$$

Take $R_2 = -\dfrac{\sin \theta}{\cos \theta -1} R_1 + R2$, which gives

$$\begin{pmatrix} \cos \theta -1 & \sin \theta \\ 0 & 0 \end{pmatrix}v_1 = 0$$

You should be able to take it from here.