Equivalence and reduction of bivariate polynomial matrices to their Smith forms

This paper is concerned with Smith forms of bivariate polynomial matrices. For a bivariate polynomial square matrix with the determinant being the product of two distinct and irreducible univariate polynomials, we prove that it is equivalent to its Smith form. We design an algorithm to reduce this class of bivariate polynomial matrices to their Smith forms, and an example is given to illustrate the algorithm. Furthermore, we extend the above class of matrices to a more general case, and derive a necessary and sufficient condition for the equivalence of a matrix and one of its all possible existing Smith forms.