A matrix theory is developed for the noncausal polyphase representation that underlies the theory of lifted filter banks and wavelet transforms. The theory presented here develops an extensive matrix algebra framework for analyzing and implementing linear phase two channel filter banks via lifting cascade schemes. Whole-sample symmetric and half-sample symmetric linear phase filter banks are characterized completely in terms of the polyphase-with-advance representation, and new proofs are given of linear phase lifting factorization theorems for these two principal classes of linear phase filter banks. The theory benefits significantly from a number of group-theoretic structures arising in the polyphase-withadvance representation and in the lifting factorization of linear phase filter banks. These results form the foundations of the lifting methodology employed in Part 2 of the ISO/IEC JPEG 2000 still image coding standard.