Equivalence and reduction of bivariate polynomial matrices to their Smith forms. (September 2023)

• Record Type: Journal Article
• Authors: Lu, Dong; Wang, Dingkang; Xiao, Fanghui; Zheng, Xiaopeng
• Is Part Of: Journal of symbolic computation. Volume 118(2023)
• Page Start: 1
• Page End: 16
• Publication Date: 2023-09
• Subjects: Bivariate polynomial matrices -- Smith forms -- Matrix equivalence -- Euclidean division -- Unimodular matrices; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2023.01.001
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: 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.