两类矩阵对角化问题的研究  

Study on Diagonalization of Two Kinds of Matrices

作　　者：徐香勤[1] XU Xiang-qin(Henan University of Animal Husbandry and Economy,Zhengzhou 450011,China)

机构地区：[1]河南牧业经济学院,河南郑州450011

出　　处：《山东农业大学学报（自然科学版）》2019年第2期350-352,共3页Journal of Shandong Agricultural University：Natural Science Edition

基　　金：河南省科技厅软科学研究计划项目(182400410584)

摘　　要：矩阵对角化在高等代数中占据着重要位置,成为求解矩阵论问题的必需工具。本文针对矩阵对角化的理论问题,利用Smith标准型,刻划了一类数字特殊分块矩阵可对角化的若干性质,并将结论推广到一般的域F上。同时讨论了方阵族可同时对角化的充要条件以及正规矩阵族可同时酉对角化的充要条件。实例结果表明:两个对称正定(半正定)Hermit矩阵充要条件必要性是显然的,幂零阵A的所有特征值都为零,A+B和B有相同特征值。Matrix diagonalization plays an important role in Higher Algebra and becomes a necessary tool for solving matrix theory problems.In this paper,some properties of the diagonalization of a class of special block matrices are characterized by using Smith's canonical form,and the conclusion is extended to the general field F.At the same time,the necessary and sufficient conditions for the simultaneous diagonalization of square matrices and the simultaneous unitary diagonalization of normal matrices are discussed.The results show that the necessary and sufficient conditions for two symmetric positive definite(semi-positive definite)Hermit matrices are obvious.All eigenvalues of nilpotent matrix A are zero,and A+B and B have the same eigenvalues.

关 键 词：极小多项式 Simth标准型 同时对角化 特征值 

分 类 号：O151.21[理学—数学]