  We derive spin operator matrix elements between general eigenstates of the
superintegrable Z_N-symmetric chiral Potts quantum chain of finite length. Our
starting point is the extended Onsager algebra recently proposed by R.Baxter.
For each pair of spaces (Onsager sectors) of the irreducible representations of
the Onsager algebra, we calculate the spin matrix elements between the
eigenstates of the Hamiltonian of the quantum chain in factorized form, up to
an overall scalar factor. This factor is known for the ground state Onsager
sectors. For the matrix elements between the ground states of these sectors we
perform the thermodynamic limit and obtain the formula for the order
parameters. For the Ising quantum chain in a transverse field (N=2 case) the
factorized form for the matrix elements coincides with the corresponding
expressions obtained recently by the Separation of Variables Method.
