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 $\mathbf A\in\mathbb K^{m\times n}$ (where $\mathbb K=\mathbb R$ or $\mathbb C)$ in case there is no singular gap, i.e. if $\sigma_h=\sigma_{h+1}$ (where $\sigma_1\geq \ldots\geq \sigma_p\geq 0$ denote the singular values of $\mathbf A$, 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 $\mathbf X\in\mathbb K^{n\times r}$ with $r\geq h$ 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.