Summary: By applying the transfer form of the singular value decomposition (SVD) over quaternion matrices, the relational conclusion about the rank of the matrix over a skew field and the particular skills of a partitioned matrix, the decomposition theory of quaternion matrices, which has the same numbers of rows or columns, is obtained. The form is the canonical form of quasi-diagonal matrix, which is more meaningful. The conclusions of this paper improve and generalize the corresponding results over the field of real numbers, the field of complex numbers and the quaternion field.