Abstract: In an axially symmetric magnetic field, charged particles move along the axis and rotate around the axis at the same time. Accordingly, the transverse motion can be decomposed into two parts, the focusing movement and the rotation of the meridian plane, and the 4-dimensional transfer matrix can be expressed as the product of a focusing matrix and two rotation matrices. The overall transfer matrix is re-derived in this article with the rotation matrices and the focusing matrix. Then the corresponding sigma matrix is given to further estimate the beam emittance extracted from electron cyclotron resonance ion source. The second moment of the beam distribution function is discussed in detail.