矩阵多项式环的广义Cayley-Hamilton定理  

Generalized Cayley-Hamilton Theorem for Ring of Polynomial Matrix

作　　者：邓勇[1] 

机构地区：[1]喀什师范学院数学系,新疆喀什844006

出　　处：《中央民族大学学报（自然科学版）》2014年第4期18-23,共6页Journal of Minzu University of China(Natural Sciences Edition)

摘　　要：Cayley-Hamilton定理是线性代数中一个著名的经典结果.目前,Cayley-Hamilton定理已被推广到长方形和分块矩阵,以及2维和n(n≥3)维线性系统等数学对象之上,并发挥着重要作用.可见,进一步将该定理推广到其他代数结构上有着理论和现实意义.为此,对任意数域上的方阵A,在通常的Cayley-Hamilton定理中,用det f(x)替换其特征多项式det(A-x I),其中f(x)是矩阵A的任意多项式,从而得到了矩阵多项式环中的广义Cayley-Hamilton定理.Cayley-Hamilton theorem is a famous and classic theorem of linear algebra. Currently, the theorem has been extended to rectangular matrix, block matrix, 2 - D and n - D linear systems of mathematical objects, etc. Thus, further study extending of Cayley-Hamihon theorem in areas of mathematics will have theoretical and practical significance. In this papers We replace the Cayley-Hamilton theorem det （A-xI） with detf （x） for any square matrix A, wheref （x） is any polynomial matrix, then we obtain a generalized Cayley-Hamihon Theorem in polynomial matrix ring.

关 键 词：CAYLEY-HAMILTON定理 矩阵多项式 特征多项式 x-矩阵 环 

分 类 号：O151.21[理学—数学]