Further results on the factorization and equivalence for multivariate polynomial matrices

This paper is concerned with the factorization and equivalence problems of multivariate polynomial matrices. We present a new criterion for the existence of matrix factorizations for a class of multivariate polynomial matrices, and prove that these matrix factorizations are unique. Based on this new criterion and the constructive proof process, we give an algorithm to compute a matrix factorization of a multivariate polynomial matrix. After that, we put forward a sufficient and necessary condition for the equivalence of square polynomial matrices: a square polynomial matrix is equivalent to a diagonal triangle if it satisfies the condition. An illustrative example is given to show the effectiveness of the matrix equivalence theorem.