Mx2 and My2 are provided for characterizing and measuring the beam quality of an astigmatic elliptical Gaussian beam. The value undergoes certain changes when the beam is not at an arbitrary azimuth angle . Accordingly, problem arises: merely using Mx2 and My2 cannot characterize beam quality effectively. So in this paper M2 factor matrix is introduced. The M2 factor matrix for the beam at an azimuth angle is theoretically derived. It is found that elements of the matrix for the original field are related to that beam at azimuth angle . The matrix elements, including diagonal and off-diagonal elements, versus azimuth angle have been presented on the basis of calculation results. Theoretical results are in agreement with experimental ones. Results imply that the sum of Mx2 and My2 for an astigmatic elliptical Gaussian beam is to be minimum as the principal axes correspond to the laboratory coordinate axes; off-diagonal elements, however, vary periodically with the variation in , and are to be zero as the principal axes correspond to the laboratory coordinate axes.