Let P be a projector.
Show that if then P is an orthogonal projection.
I know that for any projection P, , and that if P is an orthogonal projection, then , but I am not sure how to prove the other way around.
If P is not orthogonal then , the orthogonal complement of the kernel of P, is not equal to the range of P. Also, those two spaces have the same dimension. So there must exist a vector such that . Then . But so it is orthogonal to x. It follows (Pythagoras' theorem) that . Since , that tells you that and hence .
The subspaces and have the same dimension because they both have as a complementary subspace. If then (because they have the same dimension) these two subspaces would have to coincide, which would imply that P is orthogonal. Since P is not orthogonal, there must be an element such that . Then .