Request for Guidance about the sign of Singular Value Decomposition

Dear Friends, Kindly guide me about the sign of U and V in SVD.Suppose I have a matrix A=[3 1 1;-1 3 1] Whenever,I run the command [U,S,V]=svd(A) I get the following answers: U =[-0.7071 -0.7071;-0.7071 0.7071]

S =[3.4641 0 0;0 3.1623 0]

V =[-0.4082 -0.8944 -0.1826;-0.8165 0.4472 -0.3651;-0.4082 0.0000 0.9129]

The signs of S is ok, but all the signs of U and V are opposite against the answer written in article. Meaning that if I multiply matrix U and V with -ve sign then my answers are matching. Kindly guide me its reasons. I will be very thankful to you.

Best Regards.

Answer 1

[Kaushik Lakshminarasimhan]

The sign ambiguity of SVD output is well known. SVD does not have a unique answer because changing the signs of the corresponding vectors in U and V does not change the reconstructed output A = U*S*V'. So the answer that MATLAB gives you is not wrong.

However if you already knew all of this and you just want to know how MATLAB chooses the sign, I think it has to do with the way MATLAB implements the SVD algorithm. May be someone else can give a more elaborate reason.

[Atif Khan]

Dear Kaushik Lakshminarasimhan,

Many thanks for your reply.I am reading the following article. http://www.sandia.gov/~tgkolda/pubs/pubfiles/SAND2007-6422.pdf