Unitary and non-unitary operations on the Poincar\'e sphere and Pancharatnam-Berry phase with $\mathbf{Z}$ matrices

In polarization optics unitary and non-unitary operations can be carried out by the Jones matrix. $\mathbf{Z}$ matrix is the $4\times 4$ analogue of the Jones matrix and the Mueller matrix of a nondepolarizing optical medium can be written as $\mathbf{M}=\mathbf{Z}\mathbf{Z}^*$. Jones matrix acts on the two component complex Jones vector, while the $\mathbf{Z}$ matrix acts on the four component real Stokes vector. Polarizer and retarder $\mathbf{Z}$ matrices can be written in compact forms in terms of the components of the position vector on the Poincar\'e sphere. In this note it is shown that the Pancharatnam-Berry geometric phase can be demonstrated by unitary and non-unitary $\mathbf{Z}$ matrix operations.