The Research In Factorization Of Matrix On The Unique Factorization Ring

The research in factorization of matrix on the unique factorization ring plays an important role in symbolic Computation and Cybernetics, engineering computing network coding, circuits, signal processing, multidimensional system and so on. The matrix factorization problem on the ring is closely related to the Hermite nature of the matrix.we study the Hermite nature of the unqiue factorization ring problem,we also study matrix decomposition on the unique matrix decomposition and have made some preliminary results. Firstly,we drawn that the two elements is prime b,v, if c∈R, b|(vc) then b|c, if M=ρ(G),obtained, M have nuclear representation. Secondly,There is one important significant results:For the K-Hermite ring R, A∈Rl×n(l≤n), rank(A)≥l-1, d is all-level subtypeany great common factor of A,then A can be embedded into a matrix(A N), and det(A N)=d. Thirdly, For the d-Hermite ring, F∈Rl×m(l<m) is a ZLP matrix,then it can be embed into an m×m reversible matrix, these conclusions about matrix decomposition studies lay the foundation for this purpose on the decomposition ring.Predecessors have studied the matrix factorization problem on multivariate polynomial ring,because multivariate polynomial ring is a special kind of unique factorization ring. For a non-regular factor whether we can draw conclusions as the multivariate polynomial ring, through efforts we cited a counterexample.Under the full rank case, about the Lin-Bose problem on he ring, Liu gives proof which is easily understood, the last of the chapter is the study of the case in non-full rank.