多项式环上零左素矩阵的研究

发布时间：2018-05-04 08:17

本文选题：多项式环 + 多元多项式矩阵　；参考：《湖南科技大学》2017年硕士论文

【摘要】：多元多项式环上的多元多项式矩阵是代数学中的重要内容,而大多数工程问题都可以转化成多元多项式矩阵来进行求解,但多元多项式矩阵的求解是多项式矩阵理论中非常困难的。目前已经有不少学者对其进行研究,并取得了许多有意义的结果,尤其是对于一元、二元多项式矩阵,问题得到了比较好的解决,但是对于多元多项式矩阵的分解以及等价问题还存在着争议。本文主要致力于研究多项式环上零左素矩阵的相关性质,进而探讨矩阵等价的相关条件。主要包括:①寻找零左素矩阵嵌入幺模矩阵的新方法;②探究幺模矩阵分解成初等矩阵的乘积的有效方法,并相应的给出了基本算法;①分别探讨了一元和多元多项式环上的多项式矩阵等价于矩阵的有效充要条件,我们的结论较大的完善了参考文献中的相关结论,为更好的解决工程问题提供了有效依据。本文共分为六个章节。第一章是绪论部分,包括历史背景和研究现状等方面。第二章基础知识部分,主要是本文中所用到的代数学中的相关知识。第三章讨论了多项式环上的零左素矩阵的相关性质,并且得出了以下结论:对任意的幺模矩阵A都可以分解成初等矩阵的乘积的形式;对任意的零左素矩阵A都能找到一个幺模矩阵F,使得A是F的前l行所构成的子矩阵;主要对上述结论进行了构造性证明并且都给出相应的算法。第四章探讨了一元多项式环上矩阵等价的条件,并得出下列结论:任意的一元多项式矩阵A等价于矩阵的充要条件是A中所有q级子式生成单位。第五章在一元多项式环上矩阵等价的基础上加以延伸,对多元多项式矩阵的等价也进行了探讨,主要得出结论如下:对任意的多元多项式矩阵A等价于矩阵的充要条件是A中行向量所生成子模的某q行所构成的矩阵M是零左素的。第六章总结与展望。
[Abstract]:The multivariate polynomial matrix over a multivariate polynomial ring is an important part of algebra, and most engineering problems can be transformed into a multivariate polynomial matrix to be solved. But the solution of multivariate polynomial matrix is very difficult in polynomial matrix theory. At present, many scholars have studied it and obtained many meaningful results, especially for monadic, binary polynomial matrices, the problem has been solved fairly well. However, the decomposition and equivalence of multivariate polynomial matrices are still controversial. This paper is devoted to the study of the properties of zero left prime matrices over polynomial rings and the relative conditions for the equivalence of matrices. This paper mainly includes the new method of finding the zero left prime matrix embedded in the unitary matrix. 2. The effective method of factorizing the unitary matrix into the product of the elementary matrix is discussed, and the basic algorithm is given accordingly. 1. The sufficient and necessary conditions for the polynomial matrix on the polynomial ring of monadic and multivariate polynomial to be equivalent to the matrix are discussed respectively. Our conclusion greatly improves the relevant conclusions in the references and provides an effective basis for better solving the engineering problems. This paper is divided into six chapters. The first chapter is the introduction, including historical background and research status. The second chapter is the basic knowledge, which is mainly the related knowledge of algebra used in this paper. In chapter 3, we discuss the properties of zero left prime matrices over polynomial rings, and draw the following conclusions: for any unitary matrix A, we can decompose it into the product of elementary matrix; An unimodular matrix F can be found for any zero left prime matrix A, such that A is a submatrix of the first line of F, and the above conclusions are proved structurally and the corresponding algorithms are given. In chapter 4, we discuss the condition of matrix equivalence over a polynomial ring, and get the following conclusion: any polynomial matrix A is equivalent to the matrix if and only if all subforms of order Q in A are generated. In chapter 5, the equivalence of multivariate polynomial matrices is also discussed by extending the equivalence of matrices over a polynomial ring. The main conclusions are as follows: for any multivariate polynomial matrix A is equivalent to the matrix if and only if the matrix M formed by a Q line of a submodule generated by a row vector in A is zero left prime. Chapter six summarizes and prospects.
【学位授予单位】：湖南科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O153.3

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