Symmetries and obtainment of Mueller-Jones matrix from output Stokes vectors

The Jones matrix, and hence the Mueller-Jones matrix can be characterized by seven real parameters: one real and three complex parameters. In principle, any Mueller matrix can be obtained from six output Stokes vectors. If the Mueller matrix of a deterministic optical system has a symmetry then only two output Stokes vectors is enough to determine the remaining five real parameters. If, in addition to the symmetry, all nonzero complex parameters are real or pure imaginary then just a single output Stokes vector is enough. This note intends to demonstrate the internal consistency of the theorem that was recently put forward. The theorem states that there exists a relation between the outer product of input-output Stokes vectors and outer product of complex vectors. Complex vectors are obtained as a result of transformation of Stokes vectors by Z matrices. Z matrices are 4x4 analogues of Jones matrices. It is shown that determination of all relevant parameters that characterize a deterministic optical system can also be accomplished by means of the theorem. It is also discussed that complex parameters can be altered by a rotation about the z-axis, and it is shown that in some cases it is possible to make one complex parameter zero by simply rotating the optical system