If we have a projection T on a finite-dimensional inner product V, and we know that ||Tx|| = ||x||, can we conclude that T is an orthogonal projection?
The equality with the norms is enough to guarantee orthogonality of T, so I guess what I'm asking is: what is the difference between an orthogonal operator and an orthogonal projection?
On the same subject, how would a proof of the converse go? If we have an orthogonal projection T, how would we prove that ||Tx|| <= ||x||?
My idea for this was to start with 0 <= ||Tx||^2 = < Tx, Tx > = < x, Tx > and then look at this last inner product in terms of an orthonormal basis of eigenvectors of T (we know this exists since T is normal). I got stuck at this point though; I couldn't argue that the equality becomes an inequality after you pull out all the eigenvalues. Suggestions?
Any help is appreciated, thanks!