The degree of polarization is employed as a criterion to find the nine independent relations among the elements of the Mueller–Jones matrix. This procedure is applied by considering a previously determined, physically realizable Mueller matrix. On the other hand, the nine bilinear constrains are obtained by directly measuring the degree of polarization from an outgoing beam of light from an optical system by considering nine incident states of light taken from the Poincaré sphere. For practical purposes, all the incident polarization states must be scanned from the Poincaré sphere in order to satisfy the overpolarization and the overgain conditions, respectively, for the physical realizability of the Mueller matrix.