矩阵Jordan标准型在矩阵分析中的作用探讨
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摘要
矩阵的标准型理论是矩阵论中重要的一个方面,尤其关于化矩阵为Jordan标准型的理论及方法,已经列为线性微分方程组理论的必不可少的基础知识。矩阵Jordan标准型的计算是求解特征值问题的进一步发展,回顾高等代数学过的Jordan标准型,最常见的求法是通过初等因子和Jordan型中Jordan块的关系求解。文章由定义出发,介绍了Jordan标准型的求法,此外,还讨论了Jordan标准型在"矩阵分解论"、"计算矩阵多项式"和求解线性微分方程组中的应用,并通过一些例题来说明,从而感受Jordan标准型在代数学中广泛的应用价值。
The standard theory of the matrix is an important aspect in the matrix theory,especially about the theory and method of the matrix of the Jordan standard already listed as necessary basic knowledge of the theory of linear differential equations. Matrix Jordan canonical form of the calculation is applied for solution of the eigenvalue problem of further development. In view of the Jordan standard in higher algebra, the most common method is the solution through the primary factor and Jordan Jordan block in relationship. This paper starts with definition and introduces the method of Jordan standard, discusses the Jordan standard in the" matrix decomposition theory ", " computational matrix polynomials" and the application of solving linear differential equations. Then, through some examples for illustration, one can feel the Jordan standard's extensive application value in algebra.
The standard theory of the matrix is an important aspect in the matrix theory,especially about the theory and method of the matrix of the Jordan standard already listed as necessary basic knowledge of the theory of linear differential equations. Matrix Jordan canonical form of the calculation is applied for solution of the eigenvalue problem of further development. In view of the Jordan standard in higher algebra, the most common method is the solution through the primary factor and Jordan Jordan block in relationship. This paper starts with definition and introduces the method of Jordan standard, discusses the Jordan standard in the" matrix decomposition theory ", " computational matrix polynomials" and the application of solving linear differential equations. Then, through some examples for illustration, one can feel the Jordan standard's extensive application value in algebra.
引文
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[2]沈启钧.确定实非对称矩阵Jordan标准型的一种算法[J].高等学校计算机数学学报,1983(4).
[3]北京大学数学系几何与代数教研室代数小组.高等代数(第二版)[M].北京:高等教育出版社,2000.
[4]张志旭.矩阵标准型的思想及应用[J].佳木斯大学学报,2006(4).
[5]王莲花.若当标准型的计算及其应用[J].河南教育学院学报,2001(3).
[6]袁俊伟.关于用几何方法求Jordan标准型的注记[J].湖北民族学院学报,1996(2).
[7]王世超.高等代数新方法[M].徐州:中国矿业大学出版社,2003.
[8]许成亮.任意域上矩阵的Jordan标准型[J].合肥工业大学学报,2012(7).
[9]万冰蓉.矩阵若当标准型的另一证明[J].井冈山师范学院学报,2004(6).