Dominant subspace and low-rank approximations from block Krylov subspaces without a gap
In this work we obtain results related to the approximation of h-dimensional dominant subspaces and low rank approximations of matrices πβπ^mΓ n (where π=β or β) in case there is no singular gap, i.e. if Ο_h=Ο_h+1 (where Ο_1β₯β¦β₯Ο_pβ₯ 0 denote the singular values of π, and p=min{m,n}). In order to do this, we describe in a convenient way the class of h-dimensional right (respectively left) dominant subspaces. Then, we show that starting with a matrix πβπ^nΓ r with rβ₯ h satisfying a compatibility assumption with some h-dimensional right dominant subspace, block Krylov methods produce arbitrarily good approximations for both problems mentioned above. Our approach is based on recent work by Drineas, Ipsen, Kontopoulou and Magdon-Ismail on approximation of structural left dominant subspaces; but instead of exploiting a singular gap at h (which is zero in this case) we exploit the nearest existing singular gaps.
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