On The Equivalence Of The Smith Form Of 2-D Polynomial Matrices

The equivalence of the bivariate polynomial matrices can simplify many engineering calculations and be used in many fields such as circuits and physical systems and so on.Among them,the Smith forms of the bivariate polynomial matrices,because of their simple form,has occupied an important position in the research of the past few decades.However,so far,no scholar has been able to propose an easily discriminated condition and constructive method to solve the problem of the equivalence of a general bivariate polynomial matrix and its Smith form.This paper will combine the properties of bivariate polynomial rings and the existing research results to study some classes of the equivalence of the bivariate polynomial matrix and its Smith form.First,we discussed the problem of two classes of bivariate polynomial matrices to be equivalent to their Smith forms.One class is a bivariate polynomial matrix with the determinant is an irreducible polynomial p(z).We regard the matrix element on the ring(?)[z,w]as the element on the ring(?)[z][w],and use the division property of(?)[z][w]and residue class theory,it is proved that such polynomial matrix with the determinant p(z)is always equivalent to its Smith form.After that,we extend the result to the bivariate polynomial matrix with p^q(z)as its determinant,and obtain that such matrix is equivalent to their Smith forms if and only if det F(z,w)and the(l-1)×(l-1)minors have no common zero points.The other class is the bivariate polynomial matrix with(z-f(w))(w-a)as determinant,where a is a non-zero constant,and obtain that the sufficient and necessary condition of such matrix to be equivalent to their Smith forms is also det F(z,w)and the(l-1)×(l-1)minors have no common zero points.Using the properties of the ZLP matrix and the Quillen-Suslin theorem,the above conclusion is extended to the case where the determinant of the binary polynomial matrix is(z-f(w))^q₁(w-a)^q₂.Some specific examples are given to illustrate our methods.Applying above results can reduce the bivariate system matrix to a matrix with simpler form and structure,the corresponding linear system is also simplified.This paper has solved some special cases of the bivariate polynomial matrices to reduce to their Smith form.The methods are all constructive,and it can find the unimodular matrices in the unimodular transformation,which is very meaningful in practical applications.Moreover,for the equivalence of the general bivariate polynomial matrices and their Smith forms,this paper also provides new ideas and reference.