Research On The Equivalences Of Multivariate Polynomial Matrix And Multidimensional Systems

Multidimensional linear systems are usually defined by the differential or difference equations of several independent variables.The system description method based on polynomial matrix can connect the equivalence research between systems and the equivalence of polynomial matrices.Exploring the different equivalence relations between multivariate polynomial matrices can promote the study of dynamic behaviors of multidimensional systems.Based on this,the main research contents of this thesis are as follows:First,this thesis discusses how to obtain the corresponding polynomial system matrix from a given multidimensional system,and then studies several common equivalent types and the related features between the multivariate polynomial matrices and the system matrices.The relation between two types of strict system equivalence in the unvariate case is extended to the unimodular equivalence and zero coprime equivalence in the multivariate case:two multivariate polynomial matrices P1(z)? R[z]p1×q1 and P2(z)E R[z]p2×q2 are zero coprime equivalent if and only if the two matrices(?)and(?)are unimodular equivalent.The unimodular equivalence and zero prime equivalence between two multivariate polynomial matrices of the same order are considered in the case of no expansion.In addition to this,this thesis investigates the unimodular equivalence of multivariate polynomial matrices with its Smith form.Due to the complexity of the multivariate polynomial ring structure,the general multivariate polynomial matrices are not necessarily unimodular equivalent to their Smith forms.Thus,we consider a special class of multivariate polynomial matrix F(z),whose determinant is dr,d=z1-f(z2,…,zn).When d|dl-r+1(F(z),this kind of matrix is equivalent to its Smith form(?)if and only if all the(l-r)×(l-r)minors of F(z)have no common zero points.Then,the conclusion is extended to the case where the determinant is dq and dq/r|dlr+1(F(z))(q,q/r are integers).At the same time,Finally,an example is also given to illustrate our result and construction method.