Let $SO(3)$ act on the space of $3 \times 3$ real matrices by conjugation. How can I decompose the space of matrices into the sum on minimal invariant subspaces and figure out what they are isomorphic to?
I am familiar with the irreducible representations of $SU(2)$ and how they give the irreducible representations of $SO(3)$. I don't see how to relate this notion to the minimal invariant subspaces.