<p>This paper studies the equivalence theory between bivariate polynomial matrices and their Smith forms. For a class of bivariate polynomial matrices, by leveraging the special form of the greatest common divisor of the maximal minors of the matrix, the authors construct a homomorphism from the bivariate polynomial ring to a Euclidean domain. Subsequently, by applying Gaussian elimination, the matrix can be reduced to its Smith form. Consequently, the authors establish that the necessary and sufficient condition for such a type of matrix to be equivalent to its Smith form is that the reduced minors of each order generate the unit ideal.</p>