What does eig function actually do ?

I came across a statement saying that for a complex hermitian matrix,EVD is identical to SVD,where the associated right and left singular vectors are identical. I tried to verify this statement using matlab ,I used the following commands

[U1 S1]=eig(A);

[U2 S2 V2]=svd(A);

where A is hermitian matrix. i got the result as matrices S1 and S2 are identical and also U2=V2,but U1 and U2 are different,why it so?

I mean eigen decomposition says U1*S1*U1'=A SVD decomposition says U2*S2*V2'=A since U2=V2 (same as EVD decomposition) and also S1=S2,then why am i getting U1 and U2 different ??

Thanks

Answer 1

[Matt J]

Normalized eigenvectors are invariant to sign changes, so U1 and U2 could have columns differing by a sign.

Furthermore, if A has eigenvalues with multiplicity greater than 1, then there is further flexibility in the choice of eigenvectors for those eigenvalues. The space of eigenvectors then becomes multi-dimensional. You cannot be sure eig and svd will reach the same selection.

[Betha Shirisha]

@ matt thanks..

Eigen values are different and not only there is a sign varition even magnitude is also not matching..

[Matt J]

You should attach the A matrix in a .mat file so that we can try to reproduce what you are seeing.

[Betha Shirisha]

Matrix is named as Ryy instead of A

[Matt J]

Ah. In the case of complex matrices, the columns of U1 and U2 corresponding to the same eigenvalues can differ by complex scalar multiples c of amplitude abs(c)=1. Note that for any normalized complex eigenvector u, belonging to eigenvector lambda, the vector c*u is also a normalized complex eigenvector associated with lambda.

You will see that this is the case for your U1,U2 if you compute the matrix

    >> C=U1\U2

    C =

      -0.0000 + 0.0000i   0.0000 - 0.0000i  -0.0000 - 0.0000i   0.6120 - 0.7909i
       0.0000 + 0.0000i   0.0000 - 0.0000i  -0.9679 + 0.2513i   0.0000 + 0.0000i
       0.0000 + 0.0000i  -0.7120 + 0.7022i  -0.0000 + 0.0000i   0.0000 - 0.0000i
      -0.9022 + 0.4314i   0.0000 + 0.0000i  -0.0000 + 0.0000i  -0.0000 - 0.0000i