A unitary transformation is one for which U^†U^=1.
(a) Show that unitary transformations preserve inner products, in the sense that ⟨U^α∣U^β⟩=⟨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.