Let $\scrM$ be a $p$-dimensional subspace of $\scrH$ and $\scrN$ its orthogonal complement. Choosing $j$ vectors from $\scrM$ and $k-j$ vectors from $\scrN$ and forming the linear span of the antisymmetric tensor products of all such vectors, we get different subspaces of $\wedge^k\scrH$; for example, one of those is $\vee^k\scrM$. Determine all the subspaces thus obtained and their dimensionalities. Do the same for $\vee^k\scrH$.

Solution. (1). Let $e_1,\cdots,e_p$ be the orthonormal basis of $\scrM$, and $e_{p+1},\cdots,e_k$ be the orthonormal basis of $\scrN$. Then for $0\leq j\leq k$, the subspace we consider has a basis $$\bex e_{i_1}\wedge \cdots \wedge e_{i_j}\wedge e_{i_{j+1}}\wedge\cdots \wedge e_{i_k}, \eex$$ where $$\bex 1\leq i_1<\cdots<i_j\leq p<p+1\leq i_{j+1}<\cdots<i_k\leq n. \eex$$ Thus its dimension is $$\bex \sex{p\atop j}\cdot \sex{n-p\atop k-j}. \eex$$ (2). Now we consider the subspace of $\vee^k\scrH$. In this case, it has a basis $$\bex e_{i_1}\vee \cdots \vee e_{i_j}\vee e_{i_{j+1}}\vee \cdots \vee e_{i_k}, \eex$$ where $$\bex 1\leq i_1\leq\cdots\leq i_j\leq p<p+1\leq i_{j+1}\leq\cdots\leq i_k\leq n. \eex$$ Thus its dimension is $$\bex \sex{p+j-1\atop j}\cdot \sex{n-p+k-j+1\atop k-j}. \eex$$

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.3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. WebApi2 jsonp跨域问题

    一:故事背景     以前在写WebApi2的时候,一直是用作前后端分离(WebApi2 +angularjs),可是最近自己在给WebApp写接口的时候遇到了很多坑,总结一下就是跨域问题.而跨域问题 ...

  2. 【BZOJ2049】 [Sdoi2008]Cave 洞穴勘测

    Description 辉辉热衷于洞穴勘测.某天,他按照地图来到了一片被标记为JSZX的洞穴群地区.经过初步勘测,辉辉发现这片区域由n个洞穴(分别编号为1到n)以及若干通道组成,并且每条通道连接了恰好 ...

  3. 对.net orm工具Dapper在多数据库方面的优化

    Dapper是近2年异军突起的新ORM工具,它有ado.net般的高性能又有反射映射实体的灵活性,非常适合喜欢原生sql的程序员使用,而且它源码很小,十分轻便.我写本博客的目的不是为了介绍Dapper ...

  4. C# SqlConnection

    public static ArrayList Connect(string connectionString, string commandText) { ArrayList retValList ...

  5. VS2012编译出来的程序,在XP上运行,出现“.exe 不是有效的 win32 应用程序” “not a valid win32 application”

    升级vs2010到vs2012,突然发现build出来的应用程序无法运行,提示“不是有效的 win32 应用程序” or “not a valid win32 application”. 参考CSDN ...

  6. IntelliJ Idea12 破解码与中文乱码配置

    user name:JavaDeveloper serial number:92547-KY2BB-QZ0S1-PEZCV-HUT8Q-6RYY4        会出现Ok可以点击就会将软件 安装后, ...

  7. CVE爬虫抓取漏洞URL

    String url1="http://www.cnnvd.org.cn/vulnerability/index/vulcode2/tomcat/vulcode/tomcat/cnnvdid ...

  8. P147、面试题26:复杂链表的复制

    题目:请实现ComplexListNode* Clone(ComplexListNode* pHead),复制一个复杂链表.在复杂链表中,每个结点除了有一个m_pNext指针指向下一个结点外,还有一个 ...

  9. bugumongo--ConnectToMongoDB

    连接MongoDB 在能够对MongDB进行操作之前,需要使用BuguConnection连接到MongoDB数据库.代码如下: BuguConnection conn = BuguConnectio ...

  10. 【HDOJ】3727 Jewel

    静态区间第K大值.主席树和划分树都可解. /* 3727 */ #include <iostream> #include <sstream> #include <stri ...