Show that similarity preserves matrix multiplication (that is, if A^eB^e = C^e, then A_fB_f = C_f). Similarity does not, in general, preserve symmetry, reality, or hermiticity; show, however, that if S is unitary, and He is hermitian, then H^f is hermitian. Show that S carries an orthonormal basis into another orthonormal basis if and only if it is unitary.