反对合矩阵的相似对角化  

On Problem for Similar Diagonalization of Anti-involutory Matrices

作　　者：黄益生[1] 陈椰婷[1] 

机构地区：[1]三明学院信息科学学院,福建三明365004

出　　处：《三明学院学报》2013年第2期1-5,共5页Journal of Sanming University

基　　金：福建省教育厅高等学校教学质量工程资助项目(ZL0902/TZ(SJ))

摘　　要：通过证明在复数域上每一个反对合矩阵都可以对角化,指出了全体n阶反对合矩阵按矩阵的相似关系进行分类,一共可以分成n+1类。还证明了,在实数域上不存在奇数阶反对合矩阵,并且每一个偶数阶反对合矩阵都不可对角化,但是每一个2n阶反对合矩阵都相似于diag{J1,J2,…,Jn},这里Jk=(0 -1 1 0),k=1,2,…,n,因而全体2n阶反对合实矩阵按矩阵的相似关系进行分类,只有一种类型。同时,指出了每一个非零偶数维实线性空间上的反对合变换都有无穷多个。In this paper, we prove that every anti-involutory matrix over the field of complex numbers is diagonalizable, and point out that the whole anti-involutory complex matrices with order n can be altogether partitioned into n ＋ 1 classes in the sense of the similar relation of matrices. We also prove that there does not exist any anti-involutory matrix with odd order over the field of real numbers, and that every anti-involutory real matrix with even order is not diagonalizable, but if it is with order 2n, it is similar to the blocked diagonal matrix diag {J1,J2,…,Jn} where Jk=（0-1 1 0）,k=1,2,…,n,thus the whole anti-involutory real matrices with order 2n can be only partitioned into one class in the sense of the similar relation Of matrices. Moreover, we point out that there are infinite numbers of anti-involutory transformations on a real linear space with nonzero even dimension.

关 键 词：反对合矩阵 反对合变换 矩阵 相似关系 分类 

分 类 号：O151.21[理学—数学]