幂零矩阵的性质与应用

目 录

摘要 ·························································································································· 1 Abstract ······················································································································ 1 1 引言 ······················································································································· 2 2 预备知识 ················································································································· 2

2.1 概念 ·············································································································· 2 2.1 引理 ·············································································································· 3 3 幂零矩阵的性质 ········································································································ 4

3.1 幂零矩阵的特性 ······························································································· 4 3.2 矩阵是幂零矩阵的几个充分必要条件 ···································································· 6 3.3 幂零矩阵和若尔当块 ························································································· 7 3.4 幂零矩阵的其他性质 ························································································· 8 4 幂零矩阵的应用 ······································································································ 11

4.1幂零矩阵在矩阵求逆中的应用 ············································································ 11

4.1.1 可求幂零矩阵与单位矩阵和的矩阵的逆 ······················································ 11 4.1.2 求主对角线上元素完全相同的三角矩阵的逆 ················································ 13 4.2 幂零矩阵在其他方面的应用 ·············································································· 14 结论 ························································································································ 16 参考文献 ·················································································································· 16 致谢 ························································································································ 18

滁州学院本科毕业论文

幂零矩阵的性质与应用

摘要：在高等数学中，矩阵是研究和解决问题的重要工具,幂零矩阵又是一类特殊的矩阵，在矩阵

理论中具有举足轻重的地位，实际应用方面也有重要的意义。幂零矩阵具有很多好的性质，本文将深入挖掘这些性质，并且用不同的方法去分析论证这些性质。同时本文还给出幂零矩阵自身特有的一些性质，讨论了矩阵是幂零矩阵的充分必要条件，并说明其在求矩阵的逆矩阵方面的优越性，并通过例子说明其在实际中的应用。

关键词：幂零矩阵；线性变换；逆矩阵；若尔当标准型；特征值；迹.

Properties and Applications of Nilpotent Matrices

Abstract: Matrix acts as a key role in studying and solving the questions in advanced mathematics. As special

forms of matrix, nilpotent matrices play a key role not only in the theory of matrix but also in practical application. Nilpotent Matrices have many good properties. In the paper, we will find and prove with various methods these properties in profundity. The paper will give some unique properties of nilpotent matrices and discusses the necessary and sufficient condition of nilpotent matrices. Then the paper shows its superiority in solving inverse matrix, and explains its practical application by examples.

Key words: Nilpotent matrix; Linear transformation; Inverse matrix; Jordan canonical form; Characteristic;

Trace.

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滁州学院本科毕业论文

1 引言

随着科学技术的迅速发展，古典的线性代数知识已不能满足现代科技的需要，矩阵的理论和方法已成为现代科技领域必不可少的工具。诸如数值分析、微分方程、力学、网络等学科领域都与矩阵理论有着密切的联系，甚至在经济管理、金融、保险、社会科学的领域，矩阵理论也有着十分重要的作用。幂零矩阵在这些领域中也发挥着重要的作用，自20世纪60年代以来，许多学者探讨了一些幂零矩阵的性质，获得了许多重要的研究成果。1964年Give证明了n阶矩阵A是幂零矩阵的充要条件是A?0，近年来幂零矩阵得到了进一步发展。在我们学到矩阵的乘法运算时曾给出了幂零矩阵的定义，但对它的介绍甚少，因此我们将加强这方面的研究与总结。本文将归纳总结幂零矩阵的一些性质，有其自身所特有的特征，它和线性变换、若尔当标准形等方面的联系，还有其性质的具体应用，在后面的应用中我们提到了一些特殊矩阵的求逆，这体现了幂零矩阵的优越性。

邹本强，韩道兰，罗雁，黄宗文，谷国梁等在文献[1-4]中给出了幂零矩阵的一些性质，并证明了幂零矩阵的性质；姜海勤还在文献[3]中给出了对于一些特殊矩阵利用幂零矩阵的性质来简化矩阵求逆的计算，还有用幂零矩阵的特性求特殊矩阵的高次幂。胡秀玲，张秀福在文献[5]中证明了对于

kn维线性空间V ,必存在V的一组基使得由V的幂零线性变换生成的幂零代数N中任意元素在该

基下的矩阵均为严格上三角矩阵。王兆飞在文献[6]中利用幂零线性变换的概念，在一般数域上讨论了幂零线性变换一定存在一组基使其在这组基下的矩阵是若当形矩阵，从而给出幂零矩阵的若当标准形。吴险峰在文献[7]中利用幂零矩阵的特征值、特征多项式、相似性等性质，给出构建幂零矩阵的几种方法。李殿龙，隋思涟在文献[8]中证明了一般域上的2-幂零矩阵存在Jordan 标准型，并给出其明确表示；同时也证明了两个2-幂零矩阵相似的充要条件是它们的秩相等。杨浩波在文献[10]中探讨数域 K上 n?n矩阵与幂零矩阵的运算联系，还证得每个奇异方阵可写成一个幂零方阵和两个幂零方阵的积之和。

2 预备知识

2.1 概念

k定义1.设A为n阶方阵，若存在正整数k，使A?0，则A称为幂零矩阵。

k定义2.设A为幂零矩阵，满足A?0的最小正整数k称为A的幂零指数，并称A是k

次幂零矩阵。显然，n阶零矩阵是特殊的幂零矩阵，其幂零指数为1。

?a11?a1n??a11?an1?????定义3.设A??????，称A???????为A的转置，称

?a?a??a?a?nn?nn??1n?n12

滁州学院本科毕业论文

?An1??A11?????*A???Aii????????A?Ann??1n?

为A的伴随矩阵，其中Aij(i,j?1,2,??,n)为A中元素aij的代数余子式。

定义4.n阶矩阵A?(aij)，

?ai?1nii称为A的迹，记为Tr(A)。显然，A的全体特征值的和等于

Tr(A)。

定义5. 形为

??0?00???1??00??J(?,t)????????

??00??0???00?1????的矩阵称为若当块，其中?为复数，由若干个若当块组成和准对角称为若当形矩阵。

定义6.f(?)??E?A称为矩阵A的特征多项式。满足f(?)??E?A?0的?的值

称为矩阵A的特征值。

2.1 引理

引理1.设A，B为n阶方阵，则?AB???B?A?,?AB??BA。

***引理2.相似矩阵具有相同的特征值。

引理3.（哈密尔顿—凯莱定理）设A是n阶方阵，f(?)?式，则有f(A)?0。

引理4.设?1,?2,?,?n为n阶矩阵A的特征值，则有

?E?A是A的特征多项

trA??1??2????n，A??1?2????n

且对任意的多项式f(x)有f(A)的特征值为f(?1),f(?2),?,f(?n)。

引理5.每一个n阶的复矩阵A都与一若当形矩阵相似，这个若当形矩阵除去若当块的

排序外被矩阵A唯一决定的，它称为A的若当标准形。

引理6.若当形矩阵的主对角线上和元素为它的特征值。

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