# SVD on Correlation Matrix

By doing some numerical examples I have satisfied myslef that the left and right singular vectors are equal for such matrices.

http://people.revoledu.com/kardi/tutorial/LinearAlgebra/SVD.html

However I would like to see a proof of this mathematically.

Also it seems to contradict the definintion given here:

http://en.wikipedia.org/wiki/Singular_value_decomposition

which says that:

"Non-degenerate singular values always have unique left- and right-singular vectors, up to multiplication by a unit-phase factor eiφ (for the real case up to sign). Consequently, if all singular values of M are non-degenerate and non-zero, then its singular value decomposition is unique, up to multiplication of a column of U by a unit-phase factor and simultaneous multiplication of the corresponding column of V by the same unit-phase factor."

So how come for a correlation matrix the svd can have equal right and left singular vectors without becoming degenerate?

Baz