In this work we obtain results related to the approximation of $h$-dimensional dominant subspaces and low rank approximations of matrices $ A\in\mathbb K^{m\times n}$ (where $\mathbb K=\mathbb R$ or $\mathbb C)$ in case there is no singular gap at the index $h$, i.e. if $\sigma_h=\sigma_{h+1}$ (where $\sigma_1\geq \ldots\geq \sigma_p\geq 0$ denote the singular values of $ A$, and $p=\min\{m,n\}$). In order to do this, we develop a novel perspective for the convergence analysis of the classical deterministic block Krylov methods in this context. Indeed, starting with a matrix $ X\in\mathbb K^{n\times r}$ with $r\geq h$ satisfying a compatibility assumption with some $h$-dimensional right dominant subspace, we show that 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. The main difference between our work and previous work on this topic is that instead of exploiting a singular gap at $h$ (which is zero in this case) we exploit the nearest existing singular gaps.

翻译：本文针对矩阵$A\in\mathbb K^{m\times n}$（其中$\mathbb K=\mathbb R$或$\mathbb C$）在指标$h$处无奇异间隙（即$\sigma_h=\sigma_{h+1}$，其中$\sigma_1\geq \ldots\geq \sigma_p\geq 0$表示$A$的奇异值，$p=\min\{m,n\}$）的情形，获得了关于$h$维主子空间逼近与低秩近似的相关结论。为此，我们为经典确定性块Krylov方法在此背景下的收敛性分析提出了全新视角。具体而言，从满足与某$h$维右主子空间兼容性假设的矩阵$X\in\mathbb K^{n\times r}$（其中$r\geq h$）出发，我们证明了块Krylov方法能够对上述两个问题产生任意精度的近似解。本方法基于Drineas、Ipsen、Kontopoulou和Magdon-Ismail关于结构左主子空间近似的最新研究成果。与以往工作的核心区别在于：我们并未利用指标$h$处的奇异间隙（此处为零），而是利用了最近邻的现存奇异间隙。