For the matrix
$$\left(\begin{array}{cc} 2 & -1 \\ -1 & 2 \\ \end{array}\right)$$
I find $\lambda=1,3$ which gives me $\lambda_{1}=\frac{1}{\sqrt{2}}\left(\begin{array}{c} 1 \\ 1 \\ \end{array}\right)$ and $\lambda_{3}=\frac{1}{\sqrt{2}}\left(\begin{array}{c} 1 \\ -1 \\ \end{array}\right)$
Now if I make the eigenvector matrix $$C = \frac{1}{2}\left(\begin{array}{cc} 1 & 1 \\ -1 & 1 \\ \end{array}\right)$$
Then I can identify the rotation angles by setting the rotation matrix $$R = \left(\begin{array}{cc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \\ \end{array}\right)$$
However if the eigenvectors are ordered the opposite way, it doesn't make sense with the rotation matrix. (Unless this means that we are interpreting it as a passive transformation instead of an active transformation?) Is this a case where order of eigenvectors does actually matter?