Every $k\times k$ positive matrix $A=(a_{ij})$ can be realised as a Gram matrix, i.e., vectors $x_j$, $1\leq j\leq k$, can be found so that $a_{ij}=\sef{x_i,x_j}$ for all $i,j$.

Solution. By Exercise I.2.2, $A=B^*B$ for some $B$. Let $$\bex B=(x_1,\cdots,x_k). \eex$$ Then $$\bex A=\sex{\sef{x_i,x_j}}. \eex$$

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.10的更多相关文章

  1. [Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.2.10

    (1). The numerical radius defines a norm on $\scrL(\scrH)$. (2). $w(UAU^*)=w(A)$ for all $U\in \U(n) ...

  2. [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 ...

  3. [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 ...

  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 ...

随机推荐

  1. 获取网页上数据（图片、文字、视频）－b

    Demo地址:http://download.csdn.net/detail/u012881779/8831835 获取网页上所有图片.获取所有html.获取网页title.获取网页内容文字... . ...

  2. 信息化的“五观”与“N为”

    系统观:联系的总体的看 生产力工具观:人与工具关系 马克思政治经济学 辨证观:发展的看 技术观:第一生产力  改变生产生活(生存)方式 信息论观:“人”联网 控制论 工程观:群体合作的智力游戏论

  3. ubuntu下安装GTK过程

    GTK的安装过程比较繁琐,以前也安装过,但没有记录下来,Google一下记录下来备用. 利用此方法成功在ubuntu12.04下安装GTK 2.24.10 1.安装gcc/g++/gdb/make 等 ...

  4. JPA2 关于 PagingAndSortingRepository

    And --- 等价于 SQL 中的 and 关键字,比如 findByUsernameAndPassword(String user, Striang pwd): Or --- 等价于 SQL 中的 ...

  5. java、js的编码、解码

    如果在地址栏挂载参数,特别是包含中文,往往要进行编码,取值时再解码,以下是java和js中编码.解码的各自方法. java: @Test public void test3() throws Unsu ...

  6. js根据id、pid把数据转为树结构

    //格式化树数据 function toTreeData(data) { var pos = {}; var tree = []; var i = 0; while (data.length != 0 ...

  7. Android TabHost 动态修改图标或者动态改变标题

    那时客户需要实现在TabHost标题上动态显示从数据库获取的个数.起初这样思考的,从数据库 获取个数是非常简单,但是要把获取的个数显示在TabHost标题,思前想后,想用Handler来异步实现消息传 ...

  8. laravel Authentication and Security

    Creating the user modelFirst of all, we need to define the model that is going to be used to represe ...

  9. uva 11817 - Tunnelling the Earth

    题意:从地球上的一个点到另一个点,求两点的球面距离和直线距离之差.假定地球是正球体,半径为6371009米. #include<iostream> #include<cmath> ...

  10. sizeof(数组)

    这里就不讨论一般的数组长度计算了,只说明一下任何数据到了函数的形参中都将退化为指针,所以计算大小的时候,也是计算的指针的大小 直接上代码了 // class sizeof测试.cpp : 定义控制台应 ...