I am dealing with a problem similar to principal component analysis. Aka, I have a matrix and i want to recover the 'most efficient basis' to exaplin the matrix variability. With a square matrix these are the eigenvectors, weighted by the eigenvalues.
Originally, I was dealing with square matrices, and I used eigendecomposition to recover the eigenvectors, as explained above. Now however, I am dealing with rectangular matrices and using SVD to recover the efficient basis, ie A=USV' where the vectors of U are the recovered basis weighted by S, the singular values.
In my particular application, the sign of the eigenvalues/singular values makes a difference.
Here is my question: with eigendecomposition and square matrices, the eigenvalues will be positive/negative. With SVD, the singular values re contrained to be the absolute value of the eigenvalues, ie s_i = |lamba_i|.
Is there anyway to recover the 'sign' of the eigenvalues through SVD?