Abstract: An analytic solution to the Non-Negative Eigenvalue Decomposition (NNED) in the non-reflection symmetry case is derived for the first time, which is named as NNED with non-reflection symmetry. It is applied to the Freeman decomposition, and then a Freeman decomposition based on NNED with non-reflection symmetry is proposed. During the Freeman decomposition, the NNED with non-reflection symmetry is used to extract volume scattering power, and adjust volume scattering power, double-bounce scattering power and surface scattering power to ensure the remainder covariance matrix has no negative eigenvalues. Compared with the Freeman decomposition based on NNED with reflection symmetry, the proposed decomposition method can availably use the non-diagonal elements which are regarded as zeros in the reflection symmetry case, and can ensure the remainder covariance matrix has no negative eigenvalues. The real-POLSAR-data experiment shows the proposed decomposition method can markedly enhance the double-bounce scattering power and weaken the volume scattering power.