主理想环上矩阵可对角化的新判据NewDiagonalization-山东农业大学.PDFVIP

( ),2015,46(4):625-627 VOL.46 NO.4 2015 山东农业大学学报 自然科学版 Journal of Shandong Agricultural University (Natural Science Edition ) doi:10.3969/j.issn.1000-2324.2015.04.030 主理想环上矩阵可对角化的新判据 邓 勇 喀什师范学院数学系, 新疆 喀什市 844006 摘 要: 矩阵的对角化问题在矩阵理论中占有重要地位.为将域上矩阵可对角化的结果进行推广,研究了主理想环上矩 阵的可对角化问题,获得了主理想环上一类具有最小多项式m(λ)=(λ−α)(λ−β),α≠β的矩阵可对角化的充分必要条件.在此 , A,B , [A,B] 基础上进一步证明了具有二次最小多项式的两个可对角化矩阵 有公共特征向量当且仅当它们的交换子 是奇 异矩阵. 关键词: 主理想环; 对角化; 最小多项式; 特征向量; 交换子 中图法分类号:O151.21; O157.3 文献标识 :A 文章编号:1000-2324(2015)04-0625-03 NewDiagonalizationConditionforMatricesinaDomainof PrincipalIdeal DENG Yong DepartmentofMathematics/Kashgar Teachers’ College,Kashgar844006, China Abstract: The diagonalization of matrices has an important position in the matrix theory. In order to expand the results for diagonalization of matrices over fields, we discussed the diagonalization of matrices over a domain of principal ideals, and obtained the necessary and sufficient conditions of diagonalization of matrices over a domain of principal ideals with minimal polynomial m(λ)=(λ−α)(λ−β),α≠β. Further, on the basis of the obtained results, the conditions under which the matrices A and Bhave common eigenvectors if and only if their commutator [A,B] is singular matrix, was proved. Keywords: Domain of principal ideals; diagonalization; minimal polynomial; eigenvector; commutator 1 研究现状 为方便讨论,本文用 表示有单位e0 的主理想环; 表示nn阶单位矩阵; 表示 上 R I M (R) R