MATLAB eig returns inverted signs sometimes

Question

I'm trying to write a program that gets a matrix `A` of any size, and SVD decomposes it: ``` A = U * S * V' ``` 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`. Problem is: the MATLAB function `eig` sometimes returns the wrong eigenvectors. This is my code: ``` 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; en…

Accepted Answer

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

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

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'
```

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
```

EDIT:

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...

MATLAB eig returns inverted signs sometimes

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