Polynomial Matrix Factorization And Its Applications

Factorizations of multivariate polynomial matrices are very important in multivariate linear systems theory and have wide applications in systems, controls and other areas. There are three kinds of prime matrices: zero left (right) prime, minor left (right) prime and factor left (right) prime. Corresponding to these prime matrices, there are three kinds of prime factorizations: ZLP (ZRP) factorization, MLP (MRP) factorization and FLP (FRP) factorization, which have close relations with each other. In recent years, there are systematic results on the factorizations of multivariate polynomial matrices over an arbitrary field.After a survey of some basic concepts and main results on polynomial matrix theory, we study the factorizations of matrices over D[ x ] , where is an Euclidean domain. The main results are as follows.(1) Based on the work of D. C. Youla, G. Gnavi, J .P. Guiver, N. K. Bose etc., we prove the equivalence of MLP and FLP and the existence of full rank decomposition for matrices over D[ x].(2) We present a method for solving systems of homogeneous linear equa- tions with coefficients in Z[ x].(3) We prove that every non-nilpotent matrix over D[ x] is shift equivalent to some non-singular matrix.