o****u
发帖数: 1299 | 1
~~~~~~~~~~~~~~能讲得详悉一点么？给个联接什么的
eigenvalues are in general complex. they can be real in case of |
|
a****a
发帖数: 5763 | 2 一般的解hermitian matrix, numerical recipes上有一小节专门讲
这个问题,可以直接用jacobian transformation，或者用Householder
转换成三角阵然后用QL迭代
或者干脆写成 2nx2n的实matrix 求解
if C=A+iB is hermitian
(A+iB)\cdot (u+iv) =\lambda(u+iv)
then it is equivalent to write as
|A -B | \cdot |u| = \lambda |u |
|B A | |v| |v |
不太清楚解得性质如何，不过后者应该能保证eigenvalue是实数吧
具体参见Numberical Recipes third edtion page 590
I
including |
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w*******U
发帖数: 256 | 3 the fortran library 'lapack' or its C version 'lapacke' can do this job.
you can solve the eigenvalue/eigenfunction of a complex matrix with or
without hermite symmetry (the routine of the latter would run faster).
I
including |
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A***A
发帖数: 98 | 4 A small point: Page 270, Table 4, in Rushton and Jensen.
My goodness. This is already the 21st century, and we're still seeing
Kaiser's Little Jiffy floating around: Principal components - Eigenvalue > 1
- Orthogonal varimax rotation. |
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J****x
发帖数: 37 | 5 3. It is not possible, since the covariance matrix is not semi-positive-
definite (one eigenvalue -0.0487 is negatvie). |
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b***k
发帖数: 2673 | 6 也许这个程序用c实现比C++来实现要容易的多。呵呵，我是c出身的。
这个话题似乎转到了如何学习C++的问题。
以前litaihei在板上说过一个方法，我觉得很好。
就是写个例子程序来练习C++的各种简单功能
比如写个matrix class，包括
constructor (default, copy)
destructor
copy assignment
operator overloading(including +,-,*,transpose, inverse, [],etc)
member function (eigenvalue)
output/input overloading
inheritance (bandmatrix, sparse matrix)
template(generize the matrix class)
这个matrix class写完善了后，可以再编写一点小main程序来测试它。
这个过程中也可以理解const，inline，extern等等的keyword
通过这个实践，可以初步理解C++中一些最基本的但是也是最重要的概念。 |
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S*********g
发帖数: 5298 | 7 it is actually not hard to compute.
The matrix can be easily diagonalized with eigenvalues 1,-1/3,-1/3,-1/3
we can then take diag(1,-1/3,-1/3,-1/3)^7 and transform it back
is |
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j**********7
发帖数: 4 | 8 Consider a matrix $J$. every eigenvalue of $J$ has positive real part.
Now let us consider infinite product of (1-J/n). Does
infinite product converges? and how about the converge rate?
Is there anyone can give some advice? thanks! |
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p*****k
发帖数: 318 | 9 for powers of matrix, you normally want to diagonalize the matrix first:
say P'JP = diag{j1,j2,....jm}, where P'=P^(-1), and j's are the eigenvalues of J.
if i understand your question correctly, you want to compute the limit of (I-J/n)^n when n->infty (where I is the identity matrix)?
insert bunch of PP'(=I) in the product:
(I-J/n)^n = PP'(I-J/n)PP'(I-J/n)...PP'(I-J/n)PP'=P{[P'(I-J/n)P]^n}P'
=P[diag{(1-j1/n)^n,(1-j2/n)^n,...,(1-jm/n)^n}]P'
so seems to me the limit when n->infty is:
P[diag{e^(-j |
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j**********7
发帖数: 4 | 10 Many thanks! It is quite clear and correct.
You are so kind :)
eigenvalues of J.
(I-J/n)^n when n->infty (where I is the identity matrix)?
J is diagonalizable http://en.wikipedia.org/wiki/Diagonalizable, the above procedure should work, no? |
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i****e
发帖数: 78 | 11 maybe the question is (1-J/n)^m as m-> infty? then positive real parts may
be related to the convergence and rate.
eigenvalues of J.
(I-J/n)^n when n->infty (where I is the identity matrix)?
J is diagonalizable http://en.wikipedia.org/wiki/Diagonalizable, the above procedure should work, no? |
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l******i
发帖数: 1404 | 12 二阶精度一般就够了，多出的自由度不如拿去提高别的部分，例如dissipativity。看
你算法矩阵的eigenvalue是不是小于等于1去决定是不是unconditionally stable。 |
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r*******s
发帖数: 303 | 13 其实说det > 0 也不确切。
正真定义是 所有的eigenvalue 都大于0.
所以det>0不是充分条件。 |
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J****g
发帖数: 103 | 14 一样的。 等价。
半正定 <＝> 所有eigenvalue>0 <=> 所有子式的 det>0 ? |
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e*******6
发帖数: 13 | 15 请问前辈们那个matrix的eigenvalue有什么简单的办法可以直接算出来？ |
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d*****y
发帖数: 140 | 16 我怎么感觉这个问题的最优解是np-complete. 这个我再仔细想想。
不过有个找近似解方法：
let x = (x1,...,x100), x_i = 0 or 1 x_i=1 if 低i行or列属于你要找的子矩阵。
那么x 对应的子矩阵的所有元素之和可以表示为
S = x'Wx, where x'表示转置 and W是那么100的corr matrix.
问题转化为:
max S = x'Wx
subject to ,x_i=0,1 sum(x)=20.
relax 0-1限制，W的最大eigenvalue的eigenvector是实数解，所以一个近似取x的方法
是，把eigenvector里面的绝对值最大20个元素找出来，对应的row/col就组成了你要的
matrix了。note that 这个eigenvector一定是实数，而且的所有元素必定全非负或者
全部非正，所以你总能找20个最大出来。 |
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s***e
发帖数: 267 | 17 trace condition for matrix inequality is only necessary but not sufficient.
Even if trace inequality is satisfied, the matrix inequality may still fail
to hold.
E.g., max trace element <= max eigenvalue, but not vice versa. If the x's
are not bounded, then the diagonal linear equations should be parallel with
a & b. Still need more conditions to figure out the rest. |
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o****e
发帖数: 80 | 18 Matrix A has three eigenvalues lamda1, lamda2,lamda3 and three
eigenvectors v1,v2,v3. Then given a vector X, ask what is AX? (all symbols
are real numbers and vectors)
any clue? |
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j*****4
发帖数: 292 | 19 write in matrix form then use the property of eigenvalue and eigenvector. |
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J**********g
发帖数: 213 | 20 1. if X=UDU^T, where U is orthogonormal, ie, UU^T=I, D=diag[d_1,...,d_n] is
the eigenvalue matrix, then
exp(X)=Uexp(D)U^T=U diag[exp(d_1),...,exp(d_n)] U^T.
I don't know if this what they were looking for. |
|
f********k
发帖数: 136 | 21 申请的是MSc in Mathematical and Computational Finance，我知道版上还有同学想申请他们家的，所以我凭记忆列了一下我被问到的问题，供作参考。
1. What's the definition of continuity of a f(x) at x0. Given some examples,
tell if they are continous or not。（有个例子是某个函数在有理数上为0，在无
理数上为1，问连不连续，猜了连续，其实不清楚，不知道对错)
2. Matrix, eigenvalue and eigenvectors related questions. How to find limit
(1/c^n)*A^n, when n getting to infinity, if A is a symmetric matrix and the
limit exists. (这题跟他们网上的admission exercise某一题一模一样，主要就是证
明原对称矩阵跟特征值对角矩阵相似，原问题转化为对角矩阵的问题。）
3.Some ... 阅读全帖 |
|
l******n
发帖数: 9344 | 22 1. Not continuous by definition, for any Cauchy sequence x(n) to x0, f(x(n))
has to go to f(x0) if f is continuous at x0.
2. Symmetric matrix is diagonalizable, so it only depends on the relation
between c and the eigenvalues of A.
3. Simple linear algebra, you can find answers in any textbook
4. Textbook question
5. Y <=0, MTG exists;otherwise, not
不过美国学校实在太慢了，NYU说要到4月份才有消息...
想申请他们家的，所以我凭记忆列了一下我被问到的问题，供作参考。
examples,
limit
the |
|
x******a
发帖数: 6336 | 23 A可以对角化，且eigenvalue是2, 8，所以
A=O^T*diag(2, 8)*O= O^T* diag(\sqrt{2}, \sqrt{8})*O*O^T*diag(\sqrt{2},
\sqrt{8})*O， 其中O^T为O转置
所以B=O^T*diag(\sqrt{2}, \sqrt{8})*O。
忘了叫什么名字了。 |
|
k*****y
发帖数: 744 | 24 For Prob 4, consider the plane region defined by X>0 and X+Y>0，it is easy to see that the answer is 3/8.
For Prob 3, -1/3 is necessarily a lower bound for \rho to get a semi-
definite matrix, since (1+3\rho) is the eigenvalue for (1,1,1,1)^T.
Consider a regular tetrahedron inscribed in S^2. Any angle between two rays
from the center through a vertex is arccos(-1/3). This model can be realized
by a joinly normal distribution.
It can be generalized to the case of n+1. Consider a regular (n+1)-gon，... 阅读全帖 |
|
p**e
发帖数: 41 | 25 大家好， 我有3个面试的小问题， 求解：
1， 任意选择3个点在一个圆周上， 问它们连成的三角形includes圆心的概率？ （答
案是0.25, 为什么）
2，3个player玩toss a coin的游戏， 游戏规则是首先toss到head的人赢， 如果第一
个人toss了tail， then他把coin给下一个人玩， 这样循环下去一直到有人赢。 问这
三人每人的赢的概率分别是多少？（答案是 第一个人4/7 第二个人2/7 第三个人1/7
， 怎么得到的？）
3，n×n矩阵A，其对角线元素全部为1，其他元素全部为k， 求A的eigenvalue？ |
|
y*******u
发帖数: 930 | 26 第一个 注意到圆周上任意两点均可以组成一个半圆
分别画出这2点的直径
第三点只能在这2点直径的交集内部
那么概率等于第一点和第二点之间的圆心角
所以概率是（0,1/2)
注意到是等权重的，那么概率就是1/4
第二个就是个无穷等比级数啊
级数是1/8
第一个player 就是1/2+1/16+1/64
等比级数求和 1/2/(1-1/8)=4/7
第二个的首项是1/4 求和2/7
第三个1/7
大家好， 我有3个面试的小问题， 求解：
1， 任意选择3个点在一个圆周上， 问它们连成的三角形includes圆心的概率？ （答
案是0.25, 为什么）
2，3个player玩toss a coin的游戏， 游戏规则是首先toss到head的人赢， 如果第一
个人toss了tail， then他把coin给下一个人玩， 这样循环下去一直到有人赢。 问这
三人每人的赢的概率分别是多少？（答案是 第一个人4/7 第二个人2/7 第三个人1/7
， 怎么得到的？）
3，n×n矩阵A，其对角线元素全部为1，其他元素全部为k， 求A的eigenvalue？ |
|
c*****u
发帖数: 19 | 27 抛砖引玉下
1. 固定一个点A，第二个点B在圆周上均匀分布，然后确定第三个点C的范围（连接A和
圆心，B和圆心）
2. 等比数列求和
3. 只需求A+(k-1)*I_n的eigenvalue即可。。
/7 |
|
d*****o
发帖数: 34 | 28 如果你用的是eigenvalue decomposition的话，应该是对covariance matrix而不是
data matrix做的吧。 |
|
A**u
发帖数: 2458 | 29 n×n矩阵A，其对角线元素全部为1，其他元素全部为k， 求A的eigenvalue
非常感谢 |
|
R**T
发帖数: 784 | 30 一般来说没啥关系吧，不过这里的两个矩阵比较特殊
CV=cV, c是eigenvalue
(A+B)V = cV，B是单位阵的话
AV = (c-1)V，然后这里A的形式又刚好很简单 |
|
c******0
发帖数: 59 | 31 Well normally when you consider a matrix, the ith root of it is done by:
diagonalize the matrix, into P'AP, where 'means inverse
then you take the ith root of its eigenvalues, say A becomes B
Now the ith root of Matrix is P'BP. This is because P'P = I
I assume the approach is gonna be similar to this problem |
|
B**G
发帖数: 118 | 32 本人工程phd，有点cs自学背景，无金融背景
onsite一共面了10个人，每个人1个小时左右
基本每个人都是 40%聊简历和背景+40%问面试题+20%我问问题
面试题我想是根据每个人背景会不一样的
对我来说，问的主要是数学、统计、概率、算法这几块
面试前复习了点算法（leetcode都没刷完）
看了Xinfeng Zhou绿宝书（看完了）
和Mark Joshi那本Quant Job Interview（没看完）
感谢各位面试官
感谢来自quant版的信息
下面汇报下所能记住的面试题，也就是40%那部分。
1.华人MM
聊简历后，问了finite difference数值方法
问如果边界条件是给的是无穷远处的极限值，该怎么处理
后来问了Itōs lemma的基本概念
问我期权收那些变量的影响，以及随每个变量的大致趋势
2.印度小哥
问怎么validate一个binary search tree
并写pseudo code
再问怎么只用4个0以及数学运算得到24
答案： (0!+0!+0!+0!)!=24
再问什么是hash table，如何实现的
访问和插入的时间复杂度，hash ... 阅读全帖 |
|
B**G
发帖数: 118 | 33 本人工程phd，有点cs自学背景，无金融背景
onsite一共面了10个人，每个人1个小时左右
基本每个人都是 40%聊简历和背景+40%问面试题+20%我问问题
面试题我想是根据每个人背景会不一样的
对我来说，问的主要是数学、统计、概率、算法这几块
面试前复习了点算法（leetcode都没刷完）
看了Xinfeng Zhou绿宝书（看完了）
和Mark Joshi那本Quant Job Interview（没看完）
感谢各位面试官
感谢来自quant版的信息
下面汇报下所能记住的面试题，也就是40%那部分。
1.华人MM
聊简历后，问了finite difference数值方法
问如果边界条件是给的是无穷远处的极限值，该怎么处理
后来问了Itōs lemma的基本概念
问我期权收那些变量的影响，以及随每个变量的大致趋势
2.印度小哥
问怎么validate一个binary search tree
并写pseudo code
再问怎么只用4个0以及数学运算得到24
答案： (0!+0!+0!+0!)!=24
再问什么是hash table，如何实现的
访问和插入的时间复杂度，hash ... 阅读全帖 |
|
b*******4
发帖数: 65 | 34 Let X_i be the static distribution of number of mouse is block i for i=1...
16 and X be the vector.
Matrix A 16 by 16 be the transition matrix with A_ij representing the
probability that mouse in block i will transit to block j. For static
distribution, we will have
X=AX
Then the distribution will be the eigenvector of A with the corresponding
eigenvalue having one.
This is a standard control theory (linear dynamical system) problem.
一个4x4的board，t=0的时刻，每个格子里都有一只老鼠，每一秒，老鼠会随机跳到相
邻的格子，问：（1）t=1的时刻，... 阅读全帖 |
|
y**t
发帖数: 50 | 35 This matrix has nice property that
one can get its eigenvalues immediately by observation
i.e.,the matrix can be diagnolized as
R=ADA^{-1}
where D is a diagonal matrix with D=diag(a,b,...,b)
a=x+(K-1)p,b=x-p
so one can get det(R)=ab^{K-1}
and R^{-1}=AD^{-1}A^{-1} and we finally get R^{-1}
has the form
|c d d d … d|
|d c d d … d|
|..|
|……………..|
|d d d d 璫c|
where c=[x+(K-2)p]/{(x-p)[x+(K-1)p]},
d=-p/{(x-p)[x+(K-1)p]},it's easy to verify that
it it th |
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l**n
发帖数: 67 | 36
The standard way will be to solve for the secular equation for eigenvalue
and eigenfunction as described in Goldstein(1980) But you problem
seems has a strong symmetry. It's rotational inviriant along the direction
of vector a^i. So apparently, one of the principal direction will be along
that's direction (a_x,a_y,a_z) and the other two principal directions
are degenerate(any two orthogonal direction perpendicular to a^i) since you
don't have more information other than vector a
But what's inte |
|
l**n
发帖数: 67 | 37 I cannot be better than Goldstein in this case. It's in his book
'classical mechanics' section "eigenvalues of the inertia tensor"
In simple words,
In maths, what you want to solve is a similar transformation.
Under such a transformation(3-D rotation), your S tensor becomes
diagonal. To proof that this can always be done for a hermitian
tensor(matrix) needs one page. Just skip it here. And then
the problem simply reduced to find the eigenvectors and eigen-
values for the orginal tensor. You put |
|
s***e
发帖数: 911 | 38 【 以下文字转载自 Whisper 讨论区 】
【 原文由 space 所发表 】
For any vector b perpendicular to a, we have:
=a_ib_i=0(repeated index means summation)
Lets calculate the i-th component of (S.b):
(S.b)_i=2*a_i*-(Delta_{ik}*b_k)*a^2(repeated index k means summation)
So, (S.b)_1=a^2*b_1,(S.b)_2=a^2*b_2,(S.b)_3=a^2*b_3
Then we know b must be an eigenvecotr of S also, with the eigenvalue -a^2. |
|
c********s
发帖数: 123 | 39 A is a tensor, X is a vector, a is a scaler
AX = aX
find the eigenvalue a and eigenfunction X.
The primary axiles of A is the X. |
|
S*********g
发帖数: 5298 | 40 If x is A's eigenvector, i.e., Ax=ax
Then Bx saitisfies, A(Bx)=B(Ax)=a(Bx).
If x is nondegenerate, then Bx=bx, i.e., x is B's eigenvector.
If x is degenerate, then Bx=\sum c_i x_i where Ax_i=a x_i.
We can find linearly independent combinations of x_i, say X_i, that satisfies
B X_i= b_i X_i.
In rephrase, this means we can diaognalize B in the subspace
of eigenvalue a of A. |
|
y**t
发帖数: 50 | 41 usually,the norm in a vector space(euclidean norm)
means the sqrt of the sum of squares of components
and the norm of a matrix is defined as the Sup_{|v|!=0}
(|Av|/|v|)and this is in fact the eigenvalue which has the largest absolute
value among all evs.so if given 2 vectors a and b, a
means each component of a is less than b,then what you
stated is true.I guess the necessary condition is
very dependent on the vector v. |
|
s********k
发帖数: 107 | 42 Just for u to understand the result in very simple way:
Suppose a cube inflates at the constant rate, A and B are 2 opposits vertices
of this cube, and AB is parallel to x axis. Now let this cube rotates around
x axis when it is inflating. Now consider which side of this cube will be
puctuated by the +y axis at vertex A(or B), we know that it has 3
possibilities, since each vertex has 3 sides. Just call this is a measurement,
it seems that we have 3 eigenvalues, but we don't know which one befor |
|
c****u
发帖数: 584 | 43
1) Stretch exponential
2) 2-D cases, dx/dt=Ax A is 2*2 matrix. then you have
two eigenvalues
hu |
|
m***x
发帖数: 492 | 44
先求解：A |an> = an |an>
if an is not degenerated, A B |an> = B A|an>=an B|an>, then B|an> is also
eigenvector of A so B|an> is only different from |an> by a constant, so just
calculate which is the eigenvalue for B's eigenvector |an>
if an 简并(degree as s)，AB|an(i)>=BA|an(i)>=anB|an(i)> so B|an(i)> is still
eigenvector of A, B|an(i)> = \sum_{i}{ Bii'|an(i')}> where
Bii'=
solve this problem as degenerated problem, you will get answer. |
|
m**a
发帖数: 16 | 45 The number of nonlinear equations with exact analytic solution is very few.
The famous ones are
KdV equation
sine-Gordon equation
Nonlinear Schrodinger equation (Ginzburg-Landau equation)
People study the solution in the phase space and study the perturbation of
small terms. All in all, people have very limited knowledge of nonlinear
problems till now and not so much useful except for generating the figures.
For an example, in linear math, people could do Fourier transform,
eigenvalues, filterin |
|
p*******p
发帖数: 13670 | 46 for calculating the eigenvectors and eigenvalues for a matrix? I need to
calculate a lot of eigvenvectors and values and use them in my program. Is
there any standard fast algorithm I can get online? Thanks. |
|
n*s
发帖数: 752 | 47 lapack has C version. or u can call the fortran processures from C
lapack is the core of matlab and octave
if u just want eigenvalues, u can download octave, it's free and behaves
like matlab |
|
f*******n
发帖数: 588 | 48 发信人: Smille (冰冷如火), 信区: Thoughts
标 题: [转]心理学的故事--第十三章 社会心理学家
发信站: The unknown SPACE (Wed Mar 22 11:24:43 2000), 转信
心理学的故事 上页 下页 素心学苑
第十三章 社会心理学家
无人之境
问：现代心理学中什么领域极为忙碌和高产而又没有明确的身
份，甚至没有一个普遍接受的定义？
答：社会心理学。与其说它是一个领域，倒不如说它是心理学和
社会学之间的一片无人之境，它与社会学彼此重叠，还对好几种其它
的社会科学造成影响。自从社会心理学诞生之后，它的实践者们就开
始对它究竟是什么产生意见分歧。心理学家以一种方式给它定义，社
会学家以另一种方式给它定义（本书只关心社会心理学的心理学部
分），而大多数教科书编者都提出一些模糊的定义，他们什么都说，
什么也没说，希望把两种意见都综合进来，并覆盖这个领域整个的杂
凑课题。举个例子：“（社会心理学是）对影响一个人的社会行为的
个人和情境 |
|
f*******n
发帖数: 588 | 49 发信人: Smille (冰冷如火), 信区: Thoughts
标 题: 第十六章 社会
发信站: The unknown SPACE (Tue Mar 14 18:38:23 2000), 转信
第十六章 社会
社会发展的诸阶段——家庭——低级种族的道德——舆论和社
会风俗——道德的提高——复仇和司法——战争——财产——法
律——家庭的权利和义务——宗法的和军事的领袖——民族——社
会阶级——统治
在每天发表在我们文明国家报纸上的犯罪行为的报道中，常常
有这样一些说法，如粗野狂暴，野蛮残酷。这两个词在一般会话中
标志着这样一些行为：凶恶、暴虐、残忍。毫无疑问，较少文明的
人——蒙昧人和野蛮人的生活，一般地说比我们是较为凶恶、暴虐
和残忍的。但是，我们和他们之间的区别还不完全在这一点上。正
如前几章所证明的，蒙昧的和野蛮的部族常常或多或少明显表现着
文化的各 |
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f*******n
发帖数: 588 | 50 发信人: Smille (冰冷如火), 信区: Thoughts
标 题: 人类学——人及其文化研究
发信站: The unknown SPACE (Tue Mar 14 17:58:14 2000), 转信
[英]爱德华.泰勒 著
连树声 译 |
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