The polarization degree and polarization stability of the eigen state of polarization (ESOP) of the stimulated Brillouin scattering light in the sensing coil were theoretically analyzed for the stimulated Brillouin scattering fiber optic gyros (SBS FOG) based on the Jones matrix theory and first order random coupling effect between two polarization axes of the polarization maintaining fiber.The conclusions about stability of the ESOP and polarization degree of the stimulated Brillouin scattering light were obtained.