Unitary and non-unitary operations on the Poincaré sphere and Pancharatnam-Berry phase with Z matrices

Unitary and non-unitary operations can be carried out by the Jones matrix. Z matrix is the 4x4 analogue of the Jones matrix. Jones matrix acts on the two component complex Jones vector, while 
Z matrix acts on the four component real Stokes vector. Polarizer and retarder Z matrices can be written in terms of Stokes parameters that are directly related with the components of the position vector on the Poincaré sphere. In this note it is shown that the Pancharatnam-Berry geometric phase can be demonstrated by unitary and non-unitary Z matrix operations.