Perturbation theory for simultaneous bases of singular subspaces
Froilan M. Dopico
Departamento de Matematicas, Universidad Carlos III de Madrid
Abstract: It is well-known that the proper way to deal with perturbations of eigenvectors and singular vectors is to bound the distance between invariant or singular subspaces. From this point of view having close subspaces is equivalent to the existence of close bases. We show that this is no longer true for simultaneous bases of singular subspaces, i.e. for pairs of orthonormal bases of corresponding left and right singular subspaces appearing in a same singular value decomposition. A new perturbation theory for simultaneous singular bases is developed, both in the absolute and the relative setting, showing that important differences with respect to the sensitivity of singular subpaces appear only in the absolute case. These results guarantee, in particular, that, unlike classical algorithms as QR or divide and conquer, high relative accuracy algorithms for the singular value decomposition producing multiplicative backward errors do compute reliably such simultaneous bases of singular subspaces.