方阵的相似标准形  

Similar Standard Shape of Square Matrix

作　　者：王兆飞[1] 张贺[1] 梅瑞[1] 梁菊先[1] 

机构地区：[1]河北北方学院理学院数学系,河北张家口075000

出　　处：《河北北方学院学报（自然科学版）》2009年第1期4-8,共5页Journal of Hebei North University:Natural Science Edition

摘　　要：通过在一般的数域P上引入矩阵的初等因子组的概念,利用矩阵的不变因子的性质,首先给出了一般数域上两个方阵相似当且仅当它们有相同的初等因子组,定义了数域P上的n阶矩阵A的初等因子为P(λ)l=(sλ+a1λs-1+…+as)l的P-若当块,对复数域C上的n阶矩阵A的初等因子的若当块的概念作了推广.其次利用数域P上的λ-矩阵的等价性质及数域P上方阵的特征矩阵的等价标准形,给出了求数域上的特征矩阵的初等因子的方法.最后利用初等因子组给出了数域P上矩阵的P-若当标准形的概念,进而得到了一般数域P上的任何n阶矩阵A必相似与它的P-若当标准形J的结果.此结果细化了数域P上的矩阵的有理标准形.并且当数域是复数域时,P-若当标准形就是若当标准形,因此矩阵的P-若当标准形更一般.作为推论给出了实数域上方阵的相似标准形.Resting on introducing the concept of elementary factor group on the general number field P, using the characteristics of unchanged factor of matrix, firstly two square matrixes being similar on a general number field if and only if they have same elementary factor group are given, the elementary factor of n-th-order matrix A on the number field P being the P-Jordan block of P（λ）＇=（λ＇＋a1λ^r-1＋…＋as）^l is defined, and the concept of Jordan block of elementary factor of n-th-order matrix A on the complex field C is popularized. Then using the equivalence characters of ）,-matrix on number field P and equivalent standard shape of characteristic matrix of square matrix on the number field P, the methods to get the elementary factor of characteristic matrix on number field are given. At last, using elementary factor group, the concept of P-Jordan standard shape of the matrix on the number field P is given, and then the result that any n-th-order matrix A on the general number field P must be similar to its P- Jordan standard shape J is accomplished. The rational standard shape of the matrix on the number fieId P is detailed by this result. And when the number field is a complex field, P-Jordan standard shape is Jordan standard shape, thus the P-Jordan standard shape of matrix is more general. As conclusion, the similar standard shape of the square matrix on the real number field is given.

关 键 词：数域 方阵 相似 标准形 

分 类 号：O151.21[理学—数学]