For a fixed matrix A and perturbation E we develop purely deterministic bounds on how invariant subspaces of A and A+E can differ when measured by a suitable "row-wise" metric rather than via traditional norms such as two or Frobenius. Understanding perturbations of invariant subspaces with respect to such metrics is becoming increasingly important across a wide variety of applications and therefore necessitates new theoretical developments. Under minimal assumptions we develop new bounds on subspace perturbations under the two-to-infinity matrix norm and show in what settings these row-wise differences in the invariant subspaces can be significantly smaller than the two or Frobenius norm differences. We also demonstrate that the constitutive pieces of our bounds are necessary absent additional assumptions and therefore our results provide a natural starting point for further analysis of specific problems.

中文翻译：

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不变子空间扰动的统一边界

对于固定矩阵 A 和扰动 E，我们针对 A 和 A+E 的不变子空间在通过合适的“行方式”度量而不是通过传统规范（例如二或 Frobenius）进行测量时如何不同，开发了纯粹的确定性界限。了解关于这些度量的不变子空间的扰动在各种应用中变得越来越重要，因此需要新的理论发展。在最小假设下，我们在二到无穷矩阵范数下开发子空间扰动的新界限，并显示在什么设置下，不变子空间中的这些行差异可以显着小于二或 Frobenius 范数差异。