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摄动理论已得到充分研究，其主要结论后来被类比推广至埃尔米特非奇异矩阵。本文将进一步扩展若干相对摄动结果至可能奇异的埃尔米特矩阵，并建立一类适用于埃尔米特矩阵的通用相对扰动界。基于此，通过关联的Jordan-Wielandt矩阵，我们同时获得了秩亏缺$m\times n$矩阵奇异值的对应相对界。最后，从实用中的锐度与推导角度出发，讨论了所推导的主要相对界与Weyl绝对扰动界之间的比较。