I am doing the following exercise in "Representations of Compact Lie Groups" .
Ex: Consider the representation $S^1 \to O(2) \subset U(2)$ given by $$ \exp(it)\mapsto \begin{bmatrix}\cos t&-\sin t\\-\sin t&\cos t\end{bmatrix} = R(t)$$
Find the irreducible subspaces. Find a unitary matrix $A$ such that $AR(t)A^{-1}$ is diagonal and consists of $2$ one-dimensional subspaces.
My thought was to find the eigenspaces of $R(t)$, and those would be the invariant subspaces. My concern is that the second question is diagonalizing $R(t)$, which will give us the one-dimensional representations. To me, the two questions are asking the same thing. Is there something I'm not considering or are the two questions different sides of the same coin? Thanks for any insight.