Relative perturbation theory for eigenvalues of Hermitian positive definite matrices has been well-studied, and the major results were later derived analogously for Hermitian non-singular matrices. In this dissertation we extend several relative perturbation results to Hermitian matrices that are potentially singular, and also develop a general class of relative bounds for Hermitian matrices. As a result, corresponding relative bounds for singular values of rank-deficient $m\times n$ matrices are also obtained using related Jordan-Wielandt matrices. We also discuss a comparison between the main relative bound derived and the Weyl's absolute perturbation bound in terms of their sharpness and derivation in practice.

翻译：埃尔米特正定矩阵特征值的相对扰动理论已被充分研究，其主要结果随后被类似地推广至埃尔米特非奇异矩阵。本文进一步将若干相对扰动结果拓展至可能奇异的埃尔米特矩阵，并发展了一类适用于埃尔米特矩阵的广义相对界。借助相关Jordan-Wielandt矩阵，本文还获得了秩亏损的$m\times n$矩阵奇异值的相应相对界。此外，我们从尖锐性和实际推导角度，讨论了所推导的主要相对界与Weyl绝对扰动界之间的对比分析。