SVD computing different result in Matlab and OpenCV

Question

I wonder why there is sign difference in result for SVD computing in Matlab and OpenCV. I input the same matrix

          3.65E+06  -2.09E+06   0
 YY =    -2.09E+06  2.45E+06    0
           0         0          0

[U,S,V] = svd(YY);//Matlab


        -0.798728902689475  0.601691066917623   0
   V =  0.601691066917623   0.798728902689475   0
         0                  0                   1

cv::SVD::compute(YY, S, U, V);//opencv

     0.798839   -0.601544   0
V =  0.601544   0.798839    0
     0          0           1

I know that they use the same algo, why there is sign difference? Thanks

Answer 1

Which version of OpenCV are you using?

From http://code.opencv.org/issues/1498 it seems recent versions of OpenCV no longer use LAPACK to do SVD (as used by Matlab, I think). So the assumption that the same algorithm is being used might not be correct.

Of course YY=USV'

If you negate the first columns of U and V:

U(:,1)=-U(:,1);
V(:,1)=-V(:,1)

You will find USV' still equals YY. This works for your particular case because YY is symmetric (YY=YY').

Answer 2

The results of the SVD need not be unique. For example, I = UIV' for any unitary V = U. The example you give above in particular is rank deficient, so there is no reason to expect uniqueness.

Answer 3

Singular Value Decomposition is only defined up to a sign; the signs of U and V are arbitrary, and if they are different between MATLAB and OpenCV that does not indicate a problem.