We propose a new algorithm, the pseudopolar decomposition, to decompose a Jones or a Mueller-Jones matrix into a sequence of matrix factors: J≅J_RJ_DJ_1CJ_2C or M≅M_RM_DM_1CM_2C. The matrices J_R(M_R) and J_D(M_D) parameterize, respectively, the retardation and dichroic properties of J(M) in a good approximation, while J_iC(M_iC) are correction factors that arise from the noncommutativity of the polarization properties. The exponential versions of the general Jones matrix are used to demonstrate the pseudopolar decomposition and to calculate each one of the matrix factors. The decomposition preserves all the polarization properties of the system on the factorized J_R(M_R) and J_D(M_D) matrix terms. The algorithm that calculates the pseudopolar decomposition for experimentally determined Mueller matrices is presented.