Prove that for any vectors $$\bex u_1,\cdots,u_k,\quad v_1,\cdots,v_k, \eex$$ we have $$\bex |\det(\sef{u_i,v_j})|^2 \leq \det\sex{\sef{u_i,u_j}}\cdot \det \sex{\sef{v_i,v_j}}, \eex$$ $$\bex |\per(\sef{u_i,v_j})|^2 \leq \per\sex{\sef{u_i,u_j}}\cdot \per \sex{\sef{v_i,v_j}}. \eex$$

Solution. By Exercise I.5.1, $$\beex \bea |\det(\sef{u_i,v_j})|^2 &=\sev{ \sef{ u_1\wedge \cdots u_k,v_1\wedge \cdots \wedge v_k } }^2\\ &\leq \sen{ u_1\wedge \cdots \wedge u_k }^2\sen{ v_1\wedge \cdots \wedge v_k }^2\\ &=\det \sex{\sef{u_i,u_j}}\cdot \det \sex{\sef{v_i,v_j}}. \eea \eeex$$ Similarly, by Exercise I.5.5, we have $$\bex |\per(\sef{u_i,v_j})|^2 \leq \per\sex{\sef{u_i,u_j}}\cdot \per \sex{\sef{v_i,v_j}}. \eex$$

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.7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. easy ui 表单提交添加遮罩，避免数据重复提交

    如下图: //点击提交按钮保存数据 $('#btn_submit').click(function () { //增加遮罩层 $.messager.progress({ title: '温馨提示', ...

  2. ITaCS Change Password web part

    http://changepassword.codeplex.com/ A webpart is used to change your sharepoint AD password.

  3. linux whereis which

    whereis 命令只能用于程序名的搜索,而且只搜索二进制文件(参数-b).man说明文件(参数-m)和源代码文件(参数-s). [root@localhost ~]# whereis svn svn ...

  4. Xcode 合并分支报错

    原理和操作步骤见如下转载的两篇文章, 我所使用的 svn 客户端软件是 Mac 下面的 Versions.app v1.06 这个版本包含一个多人开发的bug bug 的解决方案见我之前转载的两篇文章 ...

  5. VS 2005部署应用程序提示“应用程序无法正常启动( 0x0150002)” 解决方案

    遇到这个问题,一定是缺少了CRT.MFC.ATL的DLL,不同版本的VS是不一样的.系统自带这些库的Release版,如果没有自带,打补丁就有了:系统不自带这些库的Debug版,所以Debug版的程序 ...

  6. bnu 4351 美女来找茬(水水)

    http://www.bnuoj.com/bnuoj/problem_show.php?pid=4351 [题意]:用最小的矩形框,框住像素点差超过5的点. [题解]:求坐标x,y最大最小值 [cod ...

  7. C#设计模式学习资料--外观模式

    http://www.cf17.com/html/article/172.html http://blog.csdn.net/scucj/article/details/1374657 http:// ...

  8. 结构体 typedef关键字

    1 结构体 #include <iostream> #include <cstring> using namespace std; void printBook( struct ...

  9. mysql 权限管理

     参考:    http://www.cnblogs.com/Richardzhu/p/3318595.html 一.MySQL权限简介 关于mysql的权限简单的理解就是mysql允许你做你全力以内 ...

  10. Extjs-4.2.1（二）——使用Ext.define自定义类

    鸣谢:http://www.cnblogs.com/youring2/archive/2013/08/22/3274135.html --------------------------------- ...