唯一分解环上的矩阵分解

发布时间：2018-03-25 03:10

  本文选题：K-Hermite环　切入点：唯一分解环　出处：《湖南科技大学》2014年硕士论文

【摘要】：唯一分解环上矩阵的分解问题在符号计算与控制论、网络编码、电路、信号处理、多维系统等工程计算方面起着重要的作用。许多环R上矩阵分解问题与环R是否具有Hermite性质密切相关,研究唯一分解环上矩阵分解问题时先要研究了它的Hermite性质,我们对唯一分解环的Hermite性质,唯一分解环上矩阵的分解等问题进行了一些有意义的探讨,取得了一些初步的结果。其中重要而有意义结果；1.因K-Hermite环上可能存在零因子,所以此环上矩阵分解问题的研究较为不便,本文主要根据K-Hermite环的定义得出此环上互素的两个元素b,v,对此环上任意c,若有b|(Vc),则b|c,从而得出若M=p(G),则M有核表示的充分条件。2.对于K-Hermite环R,A∈Rl×n(l≤n), rank(A)≥l-1,d是A的所有,×,级子式的任一极大公因式,则A可嵌入到矩阵(A N),且det(A N)=d。3.对于d-Hermite环R,F∈Rl×m(l≤m)是ZLP矩阵,则F可嵌入一个m×m阶可逆矩阵A中,这些结论为此环上矩阵分解的研究打下基础。 前人研究了多元(变)多项式环上矩阵分解问题,而多元多项式环是一类特殊的唯一分解环,我们探讨了对于唯一分解环,关于非正则因子是否也可以得出矩阵分解的相关结论,通过努力,举出了一个反例,同时也得到了一些其它有价值的结果。 对于Lin-Bose问题,在满秩情况下Li u给出简单易懂的证明方法,本文最后研究了在唯一分解环上非满秩情况下的Lin-Bose问题。
[Abstract]:The decomposition problems of matrices over a unique decomposition ring are symbolic computation and cybernetics, network coding, circuits, signal processing, Many matrix decomposition problems over a ring R are closely related to whether the ring R has Hermite property. The Hermite property of matrix decomposition problem over a unique factorization ring should be studied first when we study the matrix decomposition problem over a unique factorization ring. In this paper, we discuss the Hermite property of the unique decomposition ring and the decomposition of the matrix over the unique factorization ring, and obtain some preliminary results, among which the important and meaningful result is 1.Because there may be zero divisors on the K-Hermite ring, Therefore, it is inconvenient to study the matrix decomposition problem over this ring. In this paper, based on the definition of K-Hermite ring, we obtain two elements of coprime on this ring, b ~ (v). For any c on this ring, if there is b ~ (Vc), then b _ (c), we obtain the sufficient condition that M has kernel representation if M ~ (?) p ~ (1), then M is a sufficient condition of kernel representation. For K-Hermite ring R _ (1) A 鈭，

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