The quaternion method is introduced into the polarization optics systematically. Two quaternion representations of polarized light are presented which are based on the Poincare sphere representation and the Stokes parameters, respectively, and their equivalence relation is proved. The diversity of the quaternion representations of the polarized light is discussed. The quaternion representations of polarizing devices and systems are obtained. By using the quaternion representation, the equivalent theorem of polarizing system is proved, and the composition and the quaternion representations of the reduced equivalent system are deduced. The quaternion matrix calculation method for polarizing systems is presented, and the transform relations between the quaternion representations and the Mueller matrixes are obtained. Based on the conditional commutative low of the quaternion matrix multiplication, the optimization way of the quaternion matrix calculation method are discussed, and the application prospects of the algorithm are pointed.