describes an r-dimensional ellipsoid lying in R c , whose principal semi-axes are in the direction of u i with lengths of σ i for i = 1, ..., r. Therefore, multiplication by A deforms the unit hypersphere in R g 3 by first collapsing g 3 -r dimensions, then stretching/compressing it along the remaining r dimensions, then rotating it, resulting in an r-dimensional ellipsoid that lays in R c . The SVD recovers the directions of the principal semi-axes of this ellipsoid, {u i }, and their lengths, {σ i }, as well as each vector v i that is mapped to σ i u i . The vectors {u i } are orthonormal and, remarkably, so are the set of vectors {v i }.