It is inconvenience for the practical application of beam quality assessment because of the uncertainty when test beam quality factor about non-circular symmetry beam is done. The value of M2 will change if the reference frame or other experimental conditions are different because the test for M2 is based on reference frame. To have a unity of different laboratory coordinate systems of the M2, two cross-terms were introduced into present the relationship of beam spot spindle and laboratory coordinates under test. For describing the beam quality convenient, the M2 factor matrix is defined. The principle and method for test is studied, and a system to test the M2 factor matrix has been built. The law of beam width and quality factor M2 vary with the rotation of the coordinates has been studied by using two laser diodes who have the non-circular symmetry beam on the test system. Their M2 factor matrix have been tested and calculated at last.