Matrix perturbation bounds (such as Weyl and Davis-Kahan) are frequently used in many branches of mathematics. Most of the classical results in this area are optimal, in the worst case analysis. However, in modern applications, both the ground and the nose matrices frequently have extra structural properties. For instance, it is often assumed that the ground matrix is essentially low rank, and the nose matrix is random or pseudo-random. We aim to rebuild a part of perturbation theory, adapting to these modern assumptions. We will do this using a contour expansion argument, which enables us to exploit the skewness among the leading eigenvectors of the ground and the noise matrix (which is significant when the two are uncorrelated) to our advantage. In the current paper, we focus on the perturbation of eigenspaces. This helps us to introduce the arguments in the cleanest way, avoiding the more technical consideration of the general case. In applications, this case is also one of the most useful. More general results appear in a subsequent paper. Our method has led to several improvements, which have direct applications in central problems. Among others, we derive a sharp result for perturbation of a low rank matrix with random perturbation, answering an open question in this area. Next, we derive new results concerning the spike model, an important model in statistics, bridging two different directions of current research. Finally, we use our results on the perturbation of eigenspaces to derive new results concerning eigenvalues of deterministic and random matrices. In particular, we obtain new results concerning the outliers in the deformed Wigner model and the least singular value of random matrices with non-zero mean.

翻译：矩阵扰动界（如Weyl界和Davis-Kahan界）在数学的许多分支中被广泛使用。该领域的大多数经典结果在最坏情况分析下是最优的。然而，在现代应用中，基础矩阵与噪声矩阵通常具有额外的结构特性。例如，常假设基础矩阵本质上是低秩的，而噪声矩阵是随机或伪随机的。本文旨在重构部分扰动理论，以适应这些现代假设。我们将采用轮廓展开论证方法，该方法能够利用基础矩阵与噪声矩阵主导特征向量之间的偏斜性（当两者不相关时这一特性尤为显著）来获得优势。在本文中，我们专注于特征子空间的扰动分析。这有助于以最简洁的方式引入论证框架，避免一般情形下更复杂的技术考量。在实际应用中，该情形也是最具实用价值的场景之一。更一般性的结果将在后续论文中呈现。我们的方法带来了若干改进，这些改进可直接应用于核心问题。特别地，我们针对低秩矩阵在随机扰动下的扰动问题给出了精确结果，解决了该领域的一个开放性问题。其次，我们针对统计学中的重要模型——尖峰模型，提出了新的研究成果，连接了当前研究的两个不同方向。最后，我们利用特征子空间扰动的结果，推导出关于确定性与随机矩阵特征值的新结论。具体而言，我们在变形Wigner模型的异常值问题以及非零均值随机矩阵的最小奇异值问题上获得了新的研究成果。