The design of filter banks with multiple centers of symmetry is very difficult. In this paper, a space decomposition of an orthogonal projection matrix is studied. This decomposition plays a key role in a new complete factorization theory. In addition, the concept of a minimal starting block matrix is proposed and is used to establish a new factorization of a 2m-band paraunitary polyphase matrix with multiple centers of symmetry. This factorization has the completeness property. The different possible forms of the minimal starting block matrix, which lead to the different types of filter banks, are obtained. Through different combinations of minimal starting block matrices and orthogonal projection matrices, the general solutions of a 2m-band paraunitary system with multiple centers are obtained theoretically. The four-band issue is discussed in detail as an example.