Okay, today I want to nail down a lemma about the invariant subspaces (and, in particular, eigenspaces) of self-adjoint transformations. Specifically, the fact that the orthogonal complement of an invariant subspace is also invariant.
So let’s say we’ve got a subspace and its orthogonal complement . We also have a self-adjoint transformation so that for all . What we want to show is that for every , we also have
Okay, so let’s try to calculate the inner product for an arbitrary .
since is self-adjoint, is in , and is in . Then since this is zero no matter what we pick, we see that . Neat!