Suppose that $\alpha$ is an orthogonal endomorphism of the finite-dimensional real inner product space $V$. Suppose that $V$ is decomposed as a direct sum of mutually orthogonal $\alpha$-invariant subspaces. How small can these subspaces be made, and how does $\alpha$ act on them? Justify your answer.

Describe the possible matrices for $\alpha$ with respect to a suitably chosen orthonormal basis of $V$ when $\operatorname{dim} V=3$.