> > The second option is cheaper and should be more accurate, as it avoids the > cost and undesirable conditioning effect of computing *A*' * *A*. The > speedup is noticeable, but a negative side effect overwhelms the > improvement in my case. Sparsity is important in my application, and I am > using *U* to rotate a quadratic (and to reverse the rotation later on). > So I must perform multiple matrix-matrix multiplications involving *U*. > It turns out that the overall algorithm cost is *greater* with svd than > with eig due to the difference in density of *U*.
Up to a scale factor, the eigenvectors that you get from eig(A'A) should be identical to the corresponding singular vectors from svd(A), by the definition of the SVD. If you are seeing a difference in sparsity, I can think of only two explanations: 1) The differences are just roundoff errors. i.e. if you set U[abs(U) .< THRESHOLD] = 0 for some appropriate threshold, then the results will be the same. 2) You matrix has a lot of degeneracies, so the eigenvectors are not unique and different algorithms result in different choices. You could try to find a sparsifying unitary transform (google this) to try and find the sparsest eigenvectors for each eigenvalue with multiplicity > 1. Without knowing more about your matrix A, it's hard to say more.
