exercises

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.2

The elementary tensors $x\otimes \cdots \otimes x$, with all factors equal, are all in the subspace $\vee^k\scrH$. Do they span it?   Solution. Yes. Indeed, take $$\beex \bea &\quad (x+y)\o
·2015-10-21 11:30

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

Show that the inner product $$\bex \sef{x_1\wedge \cdots \wedge x_k,y_1\wedge \cdots\wedge y_k} \eex$$ is equal to the determinant of the $k\times k$ matrix $\sex{\sef{x_i,y_j}}$.   Solution. $
·2015-10-21 11:30

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.4.5

Suppose it is known that $\scrM$ is an invariant subspace for $A$. What invariant subspaces for $A\otimes A$ can be obtained from this information alone?   Solution. It is $\scrM\otimes \scrM$
·2015-10-21 11:29

[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 lexicographically ordered basis on the space of tensors, the matrix for $A\otimes B$ can be written in block form as
·2015-10-21 11:29

[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 that a vector $w$ mush satisfy in order that the bilinear functional $$\bex F(u,v)=\sef{x,u}\sef{y,v}+
·2015-10-21 11:27

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

(1). There is a natural isomorphism between the spaces $\scrH\otimes \scrH^*$ and $\scrL(\scrH,\scrK)$ in which the elementary tensor $k\otimes h^*$corresponds to the linear map that takes a vector $u
·2015-10-21 11:27

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.3.1

Let $A=A_1\oplus A_2$. Show that   (1). $W(A)$ is the convex hull of $W(A_1)$ and $W(A_2)$; i.e., the smallest convex set containing $W(A_1)\cup W(A_2)$.   (2). $$\beex \bea \sen{A}&
·2015-10-21 11:25

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.3.6

If $A$ is a contraction, show that $$\bex A^*(I-AA^*)^{1/2}=(I-A^*A)^{1/2}A^*. \eex$$ Use this to show that if $A$ is a contraction on $\scrH$, then the operators $$\bex U=\sex{\ba{cc} A&(I-AA^*)^
·2015-10-21 11:25

[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 its inverse is $$\bex \sex{\ba{cc} I&-A\\ 0&I \ea}. \eex$$ Use this to show that if $A,B$ a
·2015-10-21 11:25

[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. The set of all unitary matrices is a compact subset of all $n\times n$ matrices. These two sets are al
·2015-10-21 11:24

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

For any matrix $A$ the series $$\bex \exp A=I+A+\frac{A^2}{2!}+\cdots+\frac{A^n}{n!}+\cdots \eex$$ converges. This is called the exponential of $A$. The matrix $A$ is always invertible and $$\bex (\ex
·2015-10-21 11:24

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.2.9

(1). When $A$ is normal, the set $W(A)$ is the convex hull of the eigenvalues of $A$. For nonnormal matrices, $W(A)$ may be bigger than the convex hull of its eigenvalues. For Hermitian operators, the
·2015-10-21 11:24

[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 $U\in \U(n)$.   (3). $w(A)\leq \sen{A}\leq 2w(A)$ for all $A$.   (4). $w(A)=\sen{A}$ if
·2015-10-21 11:24

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.2.5

Show that matrices with distinct eigenvalues are dense in the space of all $n\times n$ matrices. (Use the Schur triangularisation)   Solution.  By the Schur triangularisation, for each mat
·2015-10-21 11:22

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.2.4

(1). The singular value decomposition leads tot eh polar decomposition: Every operator $A$ can be written as $A=UP$, where $U$ is unitary and $P$ is positive. In this decomposition the positive part $
·2015-10-21 11:22

[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 convergent power series. This is called the Neumann series.   Solution.  Since $\sen{A}<1$,
·2015-10-21 11:22

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.2.2

Show that the following statements are equivalent: (1). $A$ is positive. (2). $A=B^*B$ for some $B$. (3). $A=T^*T$ for some upper triangular $T$. (4). $A=T^*T$ for some upper triangular $T$ with n
·2015-10-21 11:21

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.2.3

(1). Let $\sed{A_\al}$ be a family of mutually commuting operators. Then, there exists a common Schur basis for $\sed{A_\al}$. In other words, there exists a unitary $Q$ such that $Q^*A_\al Q$ is uppe
·2015-10-21 11:21

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.2.1

For fixed basis of in $\scrH$ and $\scrK$, the matrix $A^*$ is the conjugate transpose of the matrix of $A$.   Solution.  $$\beex \bea (A^*)_{ij}&=e_i^*A^*f_j\\ &=\sef{e_i,A^*f_j}_
·2015-10-21 11:20

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.1.3

Use the QR decomposition to prove Hadamard's inequality: if $X=(x_1,\cdots,x_n)$, then $$\bex |\det X|\leq \prod_{j=1}^n \sen{x_j}. \eex$$ Equality holds here if and only if the $x_j$ are mutually ort
·2015-10-21 11:19

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.1.2

Let $X$ be nay basis of $\scrH$ and let $Y$ be the basis biorthogonal to it. Using matrix multiplication, $X$ gives a linear transformation from $\bbC^n$ to $\scrH$. The inverse of this is given by $Y
·2015-10-21 11:19

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]Contents

I find it may cost me so much time in doing such solutions to exercises and problems....I am sorry that
·2015-10-21 11:19

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.1.1

Given any $k$-tupel of linearly independent vectors $X$ as above, there exists a $k$-tuple $Y$ biorthognal to it. If $k=n$, this $Y$ is unique.   解答: Since $$\bex \rank(X^*X)=\rank(X)=k, \eex$$
·2015-10-21 11:18

[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 $x_j$, $1\leq j\leq k$, can be found so that $a_{ij}=\sef{x_i,x_j}$ for all $i,j$.   Solution. By
·2015-10-21 11:15

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]PrI.6.1

Given a basis $U=(u_1,\cdots,u_n)$ not necessarily orthonormal, in $\scrH$, how would you compute the biorthogonal basis $\sex{v_1,\cdots,v_n}$? Find a formula that expresses $\sef{v_j,x}$ for each $x
·2015-10-21 11:15

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.6

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
·2015-10-21 11:14

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.8

Prove that for any matrices $A,B$ we have $$\bex |\per (AB)|^2\leq \per (AA^*)\cdot \per (B^*B). \eex$$ (The corresponding relation for determinants is an easy equality.)   Solution. Let $$\bex
·2015-10-21 11:14

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.7

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(\se
·2015-10-21 11:14

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.9

(Schur's Theorem) If $A$ is positive, then $$\bex \per(A)\geq \det A. \eex$$   Solution. By Exercise I.2.2, $A=T^*T$ for some upper triangular $T$ with non-negative diagonals. Thus $$\beex \bea
·2015-10-21 11:14

[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 equal to the permanent of the $k\times k$ matrix $\sex{\sef{x_i,y_j}}$.   Solution. $$\beex \be
·2015-10-21 11:13

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.4

If $\dim \scrH=3$, then $\dim \otimes^3\scrH =27$, $\dim \wedge^3\scrH =1$ and $\dim \vee^3\scrH =10$. In terms of an orthonormal basis of $\scrH$, write an element of $(\wedge^3\scrH )\oplus \vee^3\s
·2015-10-21 11:13

Python自然语言处理学习笔记(59)：练习

6.10   Exercises 练习 ☼ Read up on one of the language technologies mentioned in this
·2015-10-21 11:06

Python自然语言处理学习笔记(49)： 练习

5.10   Exercises  练习 ☼ Search the web for "spoof newspaper headlines"
·2015-10-21 11:02

Python自然语言处理学习笔记(39)： 4.11 练习

4.11   Exercises 练习 ☼ Find out more about sequence objects using Python's help facility
·2015-10-21 11:59

Python自然语言处理学习笔记(28)：3.12 练习

3.12   Exercises  练习 ☼ Define a string s = 'colorless'.
·2015-10-21 11:55

C++ Exercises（二十）

1，来看下面这段代码： #include  < iostream > using   namespace  std; class  BaseClass { public :     BaseClass()     {  &n
·2015-10-21 10:54

笨方法学python，Lesson11，12，13，14

Exercises11代码# -*- coding:utf-8 -*- print "How old are you?"
好好先生_1028·2015-10-19 12:00

chapter 6 exercises

6–1.字符串.string模块中是否有一种字符串方法或者函数可以帮我鉴定一下一个字符串是否是另一个大字符串的一部分?yes,string.find() 检测str是否包含在string中，如果beg和end指定范围，则检查是否包含在指定范围内，如果是返回开始的索引值，否则返回-16–3.排序(a)输入一串数字,从大到小排列之.(b)跟a一样,不过要用字典序从大到小排列之.(a)list1=[]w
happyliferao·2015-10-12 14:00

【索引】Geometric Algorithms in 2D::Exercises: Beginner

AOAPCI:BeginningAlgorithmContests--TrainingGuide(RujiaLiu)Chapter4.Geometry::GeometricAlgorithmsin2D::Exercises
u011328934·2015-08-27 13:00

[Exercises]24 Graphical User Interfaces

ex1(空)ex2ex3ex4(空)ex5ex6ex7ex8ex9(空)ex10(空)ex11(空)ex12(空)ex13(空)ex14ex15(空)ex16(空)ex17ex18(空)ex19(空)ex20(空)ex21(空)ex22(空)ex23(空)ex24(空)ex25(空)ex26(空)ex27(空)ex28(空)ex29(空)ex30(空)ex31(空)ex32(空)ex33(空)ex
taoxiuxia·2015-08-25 17:00

Deep learning：四(logistic regression练习)

course=DeepLearning&doc=exercises/ex4/ex4.html。这里给出的训练样本的特征为80个学生的两门功课的分数，样本值为对应的同学是否允许被上大学，如果是允许
mydear_11000·2015-08-23 09:00

Deep learning：三(Multivariance Linear Regression练习)

course=DeepLearning&doc=exercises/ex3/ex3.html.其实在上一篇博文Deeplearning：二(linearregression练习)中已经简单介绍过
mydear_11000·2015-08-23 09:00

Deep learning：二(linear regression练习)

course=DeepLearning&doc=exercises/ex2/ex2.html。本题给出的是50个数据样本点，其中x为这50个小朋友到的年龄，年龄为2岁到8岁，年龄可有小数形式
mydear_11000·2015-08-23 09:00

【索引】Geometric Computations in 2D:Exercises: Beginner

AOAPCI:BeginningAlgorithmContests--TrainingGuide(RujiaLiu)Chapter4.Geometry::GeometricComputationsin2D:Exercises
u011328934·2015-08-17 21:00

【索引】Designing Efficient Algorithms：Exercises: Beginner

BeginningAlgorithmContests--TrainingGuide(RujiaLiu)Chapter1.AlgorithmDesign::DesigningEfficientAlgorithms：Exercises
kl28978113·2015-04-19 16:00

python sklearn accuracy_score

stable/modules/model_evaluation.html以及使用cross_validation：http://scikit-learn.org/stable/auto_examples/exercises
u010454729·2015-04-17 10:00

算法导论：打印回旋数

例如n=4时，输出的螺旋矩阵如下：下面给出我的代码：1packageorg.warnier.zhang.exercises; 2 3publicclassConvolution{ 4privateintn
Warnier-zhang·2015-04-13 18:00

深度学习 Deep Learning UFLDL 最新 Tutorial 学习笔记 1：Linear Regression

并且从新编排了内容及exercises。新的UFLDL网址为：http://ufldl.stan
songrotek·2014-11-19 15:00

[Erlang危机](3.4)习题

Exercises练习ReviewQuestions复习问题：[1].NamethecommonsourcesofoverloadinErlangsystems[2].Whatarethetwomainclassesofstrategiestohandleoverload
abv123456789·2014-11-06 20:00

Accelerated C++Exercises' Resolutions

前言《AccerleratedC++：PracticalProgrammingbyExample》作者:(美)AndrewKoenigBarbaraE.Moo    这是一本非常经典的C++书籍，大牛刘未鹏(http://blog.csdn.net/pongba/article/details/1930150)也在其博客对其给了很高的评价，豆瓣以及各大论坛都将此书列为C++经典书籍。但是大多数人在
u010066564·2014-10-15 20:00
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