MATLAB eig 有时会返回反转符号 [英] MATLAB eig returns inverted signs sometimes

问题描述

我正在尝试编写一个程序来获取任意大小的矩阵 A，然后 SVD 对其进行分解:

I'm trying to write a program that gets a matrix A of any size, and SVD decomposes it:

A = U * S * V'

其中A是用户输入的矩阵，U是A * A'的特征向量组成的正交矩阵，S是奇异值的对角矩阵，V是A' * A的特征向量的正交矩阵.

Where A is the matrix the user enters, U is an orthogonal matrix composes of the eigenvectors of A * A', S is a diagonal matrix of the singular values, and V is an orthogonal matrix of the eigenvectors of A' * A.

问题是:MATLAB 函数eig 有时会返回错误的特征向量.

Problem is: the MATLAB function eig sometimes returns the wrong eigenvectors.

这是我的代码:

function [U,S,V]=badsvd(A)
W=A*A';
[U,S]=eig(W);
max=0;
for i=1:size(W,1) %%sort
    for j=i:size(W,1)
        if(S(j,j)>max)
            max=S(j,j);
            temp_index=j;
        end
    end
    max=0;
    temp=S(temp_index,temp_index);
    S(temp_index,temp_index)=S(i,i);
    S(i,i)=temp;
    temp=U(:,temp_index);
    U(:,temp_index)=U(:,i);
    U(:,i)=temp;
end
W=A'*A;
[V,s]=eig(W);
max=0;
for i=1:size(W,1) %%sort
    for j=i:size(W,1)
        if(s(j,j)>max)
            max=s(j,j);
            temp_index=j;
        end
    end
    max=0;
    temp=s(temp_index,temp_index);
    s(temp_index,temp_index)=s(i,i);
    s(i,i)=temp;
    temp=V(:,temp_index);
    V(:,temp_index)=V(:,i);
    V(:,i)=temp;
end
s=sqrt(s);
end

我的代码返回正确的 s 矩阵，以及几乎"正确的 U 和 V 矩阵.但是有些列乘以-1.显然，如果 t 是一个特征向量，那么 -t 也是一个特征向量，但是符号倒置(对于某些列，不是全部)我没有得到 <代码>A = U * S * V'.

My code returns the correct s matrix, and also "nearly" correct U and V matrices. But some of the columns are multiplied by -1. obviously if t is an eigenvector, then also -t is an eigenvector, but with the signs inverted (for some of the columns, not all) I don't get A = U * S * V'.

有什么办法可以解决这个问题吗?

Is there any way to fix this?

示例:对于矩阵 A=[1,2;3,4] 我的函数返回:

Example: for the matrix A=[1,2;3,4] my function returns:

U=[0.4046,-0.9145;0.9145,0.4046]

和内置的 MATLAB svd 函数返回:

and the built-in MATLAB svd function returns:

u=[-0.4046,-0.9145;-0.9145,0.4046]

推荐答案

请注意，特征向量不是唯一的.乘以任何常数，包括 -1(它只是改变符号)，给出另一个有效的特征向量.鉴于特征向量的定义，这一点很清楚:

Note that eigenvectors are not unique. Multiplying by any constant, including -1 (which simply changes the sign), gives another valid eigenvector. This is clear given the definition of an eigenvector:

A·v = λ·v

MATLAB 选择将特征向量归一化为具有 1.0 的范数，即符号是任意的:

MATLAB chooses to normalize the eigenvectors to have a norm of 1.0, the sign is arbitrary:

对于 eig(A)，特征向量被缩放，使得每个的范数为 1.0.对于 eig(A,B)、eig(A,'nobalance') 和 eig(A,B,flag)，特征向量未标准化

For eig(A), the eigenvectors are scaled so that the norm of each is 1.0. For eig(A,B), eig(A,'nobalance'), and eig(A,B,flag), the eigenvectors are not normalized

现在，如您所知，SVD 和特征分解是相关.下面是一些代码来测试这个事实.请注意，svd 和 eig 返回结果的顺序不同(一个从高到低排序，另一个相反):

Now as you know, SVD and eigendecomposition are related. Below is some code to test this fact. Note that svd and eig return results in different order (one sorted high to low, the other in reverse):

% some random matrix
A = rand(5);

% singular value decomposition
[U,S,V] = svd(A);

% eigenvectors of A'*A are the same as the right-singular vectors
[V2,D2] = eig(A'*A);
[D2,ord] = sort(diag(D2), 'descend');
S2 = diag(sqrt(D2));
V2 = V2(:,ord);

% eigenvectors of A*A' are the same as the left-singular vectors
[U2,D2] = eig(A*A');
[D2,ord] = sort(diag(D2), 'descend');
S3 = diag(sqrt(D2));
U2 = U2(:,ord);

% check results
A
U*S*V'
U2*S2*V2'

我得到非常相似的结果(忽略轻微的浮点错误):

I get very similar results (ignoring minor floating-point errors):

>> norm(A - U*S*V')
ans =
   7.5771e-16
>> norm(A - U2*S2*V2')
ans =
   3.2841e-14

<小时>

为了得到一致的结果，人们通常采用要求每个特征向量中的第一个元素具有特定符号的约定.那样的话，如果你得到一个不遵循这个规则的特征向量，你可以将它乘以 -1 来翻转符号......

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To get consistent results, one usually adopts a convention of requiring that the first element in each eigenvector be of a certain sign. That way if you get an eigenvector that does not follow this rule, you multiply it by -1 to flip the sign...

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