Let $A$ be a nilpotent operator. Show how to obtain, from aJordan basis for $A$, aJordan basis of $\wedge^2A$.

Solution. Since $A$ is nilpotent, each eigenvalue of $A$ is zero, and thus there exists an basis $e_1,\cdot,e_n$ of $\scrH$ such that $$\bex A(e_1,\cdots,e_n)=(e_1,\cdots,e_n) \sex{\ba{cccc} 0_s&&&\\ &J_1&&\\ &&\ddots&\\ &&&J_t \ea},\quad J_{i}=\sex{\ba{cccc} 0&1&&\\ &\ddots&\ddots&\\ &&\ddots&1\\ &&&0 \ea}_{n_i\times n_i} \eex$$ with $$\bex s+\sum_{i=1}^t n_i=n. \eex$$ Hence $Ae_i=0$ for $$\bex i\in S=\sed{1\leq i\leq s+1, s+\sum_{i=1}^jn_i+1,\ j=1,\cdots,t-1}, \eex$$ and $Ae_k=0$ for $$\bex k\in T=\cup_{j=1}^t T_j,\quad T_j=\sed{s+\sum_{i=1}^{j-1}n_i+2\leq k\leq s+\sum_{i=1}^j n_i+2}. \eex$$ Thus $$\bex k\neq j,\ k,j\in T\lra 0\neq \wedge^2A(e_k\wedge e_l)=e_{k-1}\wedge e_{l-1}. \eex$$ Hence $\wedge^2 A$ has a Jordan basis $$\bex e_i\wedge e_j;(i\in S,i<j\leq n) \eex$$ $$\bex e_k\wedge e_{k+1};\quad\sex{k\in T}; \eex$$ $$\bex e_k\wedge e_{k+2};\quad\sex{k\in T}; \eex$$ $$\bex \cdots,\quad e_{s+2}\wedge e_n. \eex$$

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.6的更多相关文章

  1. [Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.4.1

    Let $x,y,z$ be linearly independent vectors in $\scrH$. Find a necessary and sufficient condition th ...

  2. [Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.3.7

    For every matrix $A$, the matrix $$\bex \sex{\ba{cc} I&A\\ 0&I \ea} \eex$$ is invertible and ...

  3. [Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.10

    Every $k\times k$ positive matrix $A=(a_{ij})$ can be realised as a Gram matrix, i.e., vectors $x_j$ ...

  4. [Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.5

    Show that the inner product $$\bex \sef{x_1\vee \cdots \vee x_k,y_1\vee \cdots\vee y_k} \eex$$ is eq ...

  5. [Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.1

    Show that the inner product $$\bex \sef{x_1\wedge \cdots \wedge x_k,y_1\wedge \cdots\wedge y_k} \eex ...

  6. [Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.4.6

    Let $A$ and $B$ be two matrices (not necessarily of the same size). Relative to the lexicographicall ...

  7. [Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.4.4

    (1). There is a natural isomorphism between the spaces $\scrH\otimes \scrH^*$ and $\scrL(\scrH,\scrK ...

  8. [Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.2.8

    For any matrix $A$ the series $$\bex \exp A=I+A+\frac{A^2}{2!}+\cdots+\frac{A^n}{n!}+\cdots \eex$$ c ...

  9. [Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.2.7

    The set of all invertible matrices is a dense open subset of the set of all $n\times n$ matrices. Th ...

  10. [Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.2.6

    If $\sen{A}<1$, then $I-A$ is invertible, and $$\bex (I-A)^{-1}=I+A+A^2+\cdots, \eex$$ aa converg ...

随机推荐

  1. ITTC数据挖掘平台介绍(五) 数据导入导出向导和报告生成

    一. 前言 经过了一个多月的努力,软件系统又添加了不少新功能.这些功能包括非常实用的数据导入导出,对触摸进行优化的画布和画笔工具,以及对一些智能分析的报告生成模块等.进一步加强了平台系统级的功能. 马 ...

  2. 掌握Thinkphp3.2.0----CURD

    TP-----CURD  create()创建数据----整理数据 在数据库添加等操作之前,我们首先需要对数据进行创建.何为数据创建,就是接受提交过来的数据,比如表单提交的 POST(默认)数据.接受 ...

  3. mysql错误

    安装mysql之后提示(ERROR 1045 (28000): Access denied for user 'root'@'localhost' (using password:错误 具体就是: 安 ...

  4. bootstrapt model 的多罩层,禁用罩层

    选项 有一些选项可以用来定制模态窗口(Modal Window)的外观和感观,它们是通过 data 属性或 JavaScript 来传递的.下表列出了这些选项: 选项名称 类型/默认值 Data 属性 ...

  5. erlang和java通信

    连接在 https://guts.me/2014/07/27/erlang-communicates-with-java/ 代码在 https://github.com/mingshun/jinter ...

  6. asp.net &lt;% %&gt;,&lt;%# %&gt;,&lt;%= %&gt;,&lt;%$ %&gt;区别大集合

    前台页面 <div><%--可以执行服务器代码,相当于在后台写代码,Render%><%=取后台变量或方法值,只能绑定客户端控件,绑定服务器控件时后来必须调用databi ...

  7. SketchMaster 隐私政策

    隐私政策 本应用尊重并保护所有使用服务用户的个人隐私权.为了给您提供更准确.更有个性化的服务,本应用会按照本隐私权政策的规定使用和披露您的个人信息.但本应用将以高度的勤勉.审慎义务对待这些信息.除本隐 ...

  8. Natas Wargame Level26 Writeup(PHP对象注入)

    源码: <?php // sry, this is ugly as hell. // cheers kaliman ;) // - morla class Logger{ private $lo ...

  9. OAuth 2 开发人员指南(Spring security oauth2)

    https://github.com/spring-projects/spring-security-oauth/blob/master/docs/oauth2.md 入门 这是支持OAuth2.0的 ...

  10. Java学习笔记42(数据库连接池 druid连接池)

    druid连接池: 是阿里的连接池,druid的稳定性及效率都很高,目前用的比较广,所以建议开发过程中尽量用druid连接池(支持国产最重要) druid连接池也需要配置文件,配置文件必须是prope ...