A unitary transformation is one for which Û+Û = 1.
(a) Show that unitary transformations preserve inner products, in the sense that (Ûα|Ûβ) = (α|β), for all vectors |α), β).
(b) Show that the eigenvalues of a unitary transformation have modulus 1.
(c) Show that the eigenvectors of a unitary transformation belonging to distinct eigenvalues are orthogonal.