When I use [U,S,V]=svd(A), I can reproduce A by U*S*V'. However, if I generate U and V by solving two eigenvalue problems [V,D]=eig(A'*A) and [U,D]=eig(A*A'), respectively, the resulting U and V may or may NOT satisfy U*S*V'=A, due possibly to the sign difference of some of the columns (eigenvectors). While the columns of the U's and V's produced by these two different methods have the same absolute values, they may have different signs, simply because if u_i is an eigenvector of A*A', so is -u_i.
But if U*S*V'~=A for some U and V, isn't the SVD theorem violated?
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