Rayleigh-Schr " o dinger perturbation theory is a well-known theory in quantum mechanics and it offers useful characterization of eigenvectors of a perturbed matrix. Suppose @math and perturbation @math are both Hermitian matrices, @math , @math are eigenvalues of @math in descending order, and @math are leading eigenvectors of @math and @math . Rayleigh-Schr " o dinger theory shows asymptotically, @math where @math . However, the asymptotic theory does not apply to larger @math ; in particular, it fails when @math . In this paper, we present a nonasymptotic theory with @math being a random matrix. We prove that, when @math and @math has independent and centered subgaussian entries above its diagonal, with high probability, for all @math simultaneously, under a condition on eigenvalues of @math that involves all gaps @math . This bound is valid, even in cases where @math . The result is optimal, except for a log term. It also leads to an improvement of Davis-Kahan theorem.
@cite_10: The classical random matrix theory is mostly focused on asymptotic spectral properties of random matrices as their dimensions grow to infinity. At the same time many recent applications from convex geometry to functional analysis to information theory operate with random matrices in fixed dimensions. This survey addresses the non-asymptotic theory of extreme singular values of random matrices with independent entries. We focus on recently developed geometric methods for estimating the hard edge of random matrices (the smallest singular value).
@cite_7: Matrix perturbation inequalities, such as Weyl's theorem (concerning the singular values) and the Davis-Kahan theorem (concerning the singular vectors), play essential roles in quantitative science; in particular, these bounds have found application in data analysis as well as related areas of engineering and computer science. In many situations, the perturbation is assumed to be random, and the original matrix has certain structural properties (such as having low rank). We show that, in this scenario, classical perturbation results, such as Weyl and Davis-Kahan, can be improved significantly. We believe many of our new bounds are close to optimal and also discuss some applications.
@cite_10 and @cite_7 studied the top singular vectors (containing eigenvectors as a special case) of a perturbed (rectangular) matrix. To compare our result with theirs, suppose the matrices under consideration are symmetric. When @math is of low-rank, and @math , and the perturbation is random, they proved upper bounds that improve over Davis-Kahan theorem (or Wedin's theorem ). Roughly speaking, they showed that when @math , with large probability, See Theorem 9 in @cite_7 . They also proved a more general theorem that requires a weaker assumption on the random model. Their bound is the same as what we obtained (see Example ), except for a log term. Yet it is not clear why their result requires @math , and how @math helps. Our nonasymptotic RS perturbation theory offers a natural explanation and generalizes their results (with the price of a stronger independence assumption).