Given an approximate eigenvector, its (standard) Rayleigh quotient and harmonic Rayleigh quotient are two well-known approximations of the corresponding eigenvalue. We propose a new type of Rayleigh quotient, the homogeneous Rayleigh quotient, and analyze its sensitivity with respect to perturbations in the eigenvector. Furthermore, we study the inverse of this homogeneous Rayleigh quotient as stepsize for the gradient method for unconstrained optimization. The notion and basic properties are also extended to the generalized eigenvalue problem.

翻译：给定一个近似特征向量，其（标准）瑞利商与调和瑞利商是相应特征值的两种经典近似。我们提出一种新型瑞利商——齐次瑞利商，并分析其对特征向量扰动的敏感性。进一步研究该齐次瑞利商的倒数作为无约束优化梯度法步长的应用。此外，我们将该概念及其基本性质推广至广义特征值问题。