For example, I have a two-dimensional rotation matrix
$$
 \begin{bmatrix}
 0.5091 & -0.8607 \\
 0.8607 & \phantom{-}0.5091
 \end{bmatrix}
$$
and I have a vector I'd like to rotate, e.g. $(1, -0.5)$.

My problem is to find an inverse of the rotation matrix so that I can later “undo” the rotation performed on the vector so that I get back the original vector.

The rotation matrix is not parametric, created via eigendecomposition, I can't use angles to easily create an inverse matrix.